Journal of Optimization Theory and Applications

, Volume 156, Issue 3, pp 650–682

On the Existence of a Classical Optimal Solution and of an Almost Strongly Optimal Solution for an Infinite-Horizon Control Problem

Open AccessArticle

DOI: 10.1007/s10957-012-0126-2

Cite this article as:
Bogusz, D. J Optim Theory Appl (2013) 156: 650. doi:10.1007/s10957-012-0126-2

Abstract

We consider an infinite-horizon optimal control problem with the cost functional described either by an integral over an unbounded interval (a Lebesgue integral) or by a limit of integrals (an improper Lebesgue integral). We prove some theorems on the existence of solutions to such problems. The proofs are based on appropriate lower closure theorems and some extensions of Olech’s theorem on the lower semicontinuity of an integral functional; these extensions cover the cases of functionals described by an integral over an unbounded interval and by a limit of integrals.

Keywords

Infinite-horizon optimal controlExistence of an optimal solutionLower closure theoremOlech’s theorem

1 Introduction

This paper is devoted to an infinite-horizon optimal control problem described by an ordinary differential equation. The cost functional is given either by an integral over an unbounded interval (a Lebesgue integral) or by a limit of integrals (an improper Lebesgue integral). The next section provides a simple example that illustrates the difference between the two functionals.

The problem corresponds to many phenomena that are of interest, for instance, in management science or economics: It can model the relationship between advertising, sales, and company profit or describe some production-inventory systems; see [1].

The first difficulty we encounter when analyzing this problem is how to define appropriately an optimal pair. The literature offers many different definitions; see, for example, [2]. We introduce two new concepts of an optimal pair: a classical optimal pair for the model with an integral over an unbounded interval and an almost strongly optimal pair for the model with a limit of integrals (an improper Lebesgue integral). Compared to known definitions, these new concepts are a more natural extension of the definition of an optimal pair for finite-horizon models. Some relation between known and new definitions is shown in Sect. 2.

Having adequately defined an optimal pair, we give some conditions that ensure the existence of an optimal pair in the class of locally absolutely continuous trajectories and measurable controls. Here we use the method presented in [3]. It is based on the concept of the modified Lagrangian, and on a suitable version of the lower closure theorem for multifunctions defined over an unbounded domain.

The lower closure theorem, for a bounded domain, can be found in [4, Theorem 10.7.i]. Some variants of this theorem have been obtained in [5] for a special form of the multifunction and in [6] where the assumptions involve some “equi-behavior” of integrals over a bounded interval. We prove some versions of this theorem—Theorem 6.1 and Theorem 6.2—for multifunctions in a general form, defined on the interval [0,∞[. Such a theorem, with slightly different assumptions, has been stated in [3] without proof. The proofs of our lower closure theorems are based on some extensions of the classical Olech’s theorem on the lower semicontinuity of an integral functional to the case of functions defined on an unbounded domain; see [7].

Our paper consists of seven main sections. In Sect. 2, we describe the model under study in detail and give an elementary example to justify this paper. Section 3 recalls some properties of locally absolutely continuous functions defined on the interval [0,∞[. Section 4 is devoted to the classical Olech’s theorem on the lower semicontinuity of an integral functional that involves an integral over a set of finite measure and some counterparts of this result for the functionals (\(J_{\smallint }\)) and (\(J_{\qopname \relax m{lim}}\)) with integrands that depend on four variables and with an integral over the interval [0,∞[. Section 5 concerns the concept of the modified Lagrangian and its basic properties. In Sect. 6, the lower closure theorems for the above-mentioned functionals are proven. In Sect. 7, theorems on the existence of an optimal solution to system (P) with the cost functional (\(J_{\smallint }\)) or (\(J_{\qopname \relax m{lim}}\)) are derived and some examples that illustrate the existence theorems are given. In Sect. 8, some optimality principles are given. These principles say that an optimal solution of the infinite-horizon optimal control problem given by (P) and (\(J_{\smallint }\)) or (P) and (\(J_{\qopname \relax m{lim}}\)) is optimal on each finite time interval, in the usual sense.

2 Motivation

Consider the infinite-horizon control system
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Figa_HTML.gif
with the cost functional
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Figb_HTML.gif
where f:[0,∞[×ℝn×ℝm→ℝn, F:[0,∞[×ℝn×ℝm→ℝ, A:[0,∞[ ⇉ℝn1, and \(U:\operatorname{Gr}A\rightrightarrows \mathbb{R}^{m}\). The set \(\operatorname{Gr}A\) is the graph of the multifunction A.

All integrals will hereafter signify Lebesgue integration.

A function g:[0,∞[ →ℝ is said to be summable iff the integrals of the positive part g+=max{g,0} and the negative part g=max{−g,0} are finite; the function g is said to be integrable iff at least one of these integrals is finite. The (proper) Lebesgue integral of g is \(\int_{0}^{\infty}g(t)\,dt:=\int_{0}^{\infty}g_{+}(t)\,dt-\int_{0}^{\infty }g_{-}(t)\,dt\) in both cases. See [8].

If a function f:[0,∞[→ℝ is summable on each interval ]0,T[ with T positive, then the improper Lebesgue integral is defined to be the limit \(\lim_{T\rightarrow\infty}\int_{0}^{T}f ( x ) \,dx\) whenever it exists. See [9, Chap. VII, Sect. 8] or [10, Chap. VIII, Sect. 8].

Using the integral \(\int_{0}^{\infty}F(t,x(t),u(t))\,dt\) makes it necessary to impose some conditions that ensure the summability of the function F(⋅,x(⋅),u(⋅)) on the unbounded interval [0,∞[. Such conditions are restrictive and not always satisfied in real-life applications; cf. Gale’s cake eating problem in [3] and [4]. It, therefore, seems reasonable (necessary) to consider another notion of optimality. To weaken the assumptions on F, consider the functional
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Figc_HTML.gif
instead of \(J_{\smallint}\). In such a case, it is enough to assume that F(⋅,x(⋅),u(⋅)) is locally summable (that is, summable on each bounded subinterval of [0,∞[) and there exists a finite limit \(\lim_{T\rightarrow\infty}\int_{0}^{T}F(t,x(t),u(t))\,dt\).

The difference between Jlim(x,u) and \(J_{\smallint}(x,u)\) can be better seen if one takes the function \(h(t)=\frac{\sin t}{t}\): the integral \(\int_{0}^{\infty}h(t)\,dt\) does not exist, yet \(\lim_{T\rightarrow\infty} \int_{0}^{T}h(t)\,dt\) exists and is equal to \(\frac{\pi}{2}\). In other words, the function h is neither summable nor integrable and despite that there the improper Lebesgue integral of this function exists.

To sum up, this paper assumes that there exists one of the integrals: Lebesgue or, at least, improper Lebesgue. For the sake of the reader’s convenience, we use distinct notation for them: \(\int_{0}^{\infty}\) for the Lebesgue integral and \(\lim_{T\rightarrow\infty}\int_{0}^{T}\) for the improper Lebesgue integral.

The monograph [3] introduces strong optimality: A pair (x,u) is called strongly optimal iff
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equa_HTML.gif
and the inequality
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equb_HTML.gif
holds true for each pair (x,u) which satisfies system (P) and is such that the functionF(⋅,x(⋅),u(⋅)) is locally summable on [0,∞[. It is easy to observe that strong optimality and the notion of optimality based on the functional (\(J_{\qopname \relax m{lim}}\)) (see Definition 7.8) are not equivalent. More precisely: Suppose that the pair (x,u) satisfying equation (P) is optimal in the sense of Definition 7.8; only if we assume that the function F(⋅,x(⋅),u(⋅)) is locally integrable on [0,∞[ and the limit \(\lim_{T\rightarrow\infty}\int_{0}^{T}F(t,x(t),u(t))\,dt\) exists, is it meaningful to speak about the truth of the inequality in the above definition. We have therefore decided to say that the pair optimal in the sense of Definition 7.8 is almost strongly optimal.

To the author’s knowledge, the definition of an optimal pair for problem (P) with the cost functional (\(J_{\smallint }\)) (i.e., a classical optimal solution) was not considered in the literature. Different interpretations of the integral \(\int_{0}^{\infty}f ( t,x ( t ) ,u ( t ) ) \,dt\) either in the Lebesgue sense or in the Riemann sense have been discussed in [11].

3 Locally Absolutely Continuous Functions

This section recalls a definition and some properties of locally absolutely continuous functions defined on the interval [0,∞[.

A function x:[0,∞[ →ℝ is called locally absolutely continuous on [0,∞[ iff the function x|[0,T] is absolutely continuous on [0,T] for each T>0.

The space of all locally absolutely continuous functions on [0,∞[ will be denoted by ACloc([0,∞[,ℝ). It follows from the integral representation of absolutely continuous functions on a bounded interval that x belongs to the space ACloc([0,∞[,ℝ) if and only if there exists a function \(l\in L_{\mathrm{loc}}^{1} ( [0,\infty[,\mathbb{R} ) \) and c∈ℝ such that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equc_HTML.gif
for t∈[0,∞[, where \(L_{\mathrm{loc}}^{1}([0,\infty[,\mathbb{R})\) is the space of locally summable functions on [0,∞[. Consequently, each function xACloc([0,∞[,ℝ) has the derivative \(\dot {x}(t)\) almost everywhere (a.e.) on [0,∞[.
We shall consider ACloc([0,∞[,ℝ) endowed with the topology generated by the family of seminorms
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equd_HTML.gif
where xACloc([0,∞[,ℝ) and q∈ℚ+ (the positive rationals). A sequence {xk}k∈ℕACloc([0,∞[,ℝ) converges to xACloc([0,∞[,ℝ) iff
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Eque_HTML.gif
for each q∈ℚ+. See [12, Theorem 1.37].

We can prove, in an elementary way,

Theorem 3.1

LetΦbe a continuous linear functional onACloc([0,∞[,ℝ). There exists a functiongL([0,∞[,ℝ) and a constantc∈ℝ such that\(g|_{ ] T_{1},\infty [ }\equiv0\)for someT1>0 and
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ1_HTML.gif
(1)
forxACloc([0,∞[,ℝ). Conversely, any functionalΦgiven by (1) is linear and continuous onACloc([0,∞[,ℝ).

The following characterization of weak convergence in ACloc([0,∞[,ℝ) results from the above theorem.

Theorem 3.2

A sequence {xk}n∈ℕis weakly convergent toxinACloc([0,∞[,ℝ) if and only if the following two conditions are satisfied:
  1. (i)

    the sequence\(\{ \dot{x}_{k}\vert _{[0,T]}\}_{k\in\mathbb{N}}\)is weakly convergent to\(\dot{x}\vert_{[0,T]}\)inL1([0,T],ℝ) for anyT>0,

     
  2. (ii)

    the sequence {xk(0)}k∈ℕis convergent tox(0) in ℝ.

     

This theorem implies the following two results:

Theorem 3.3

If a sequence {xk}k∈ℕis weakly convergent toxinACloc([0,∞[,ℝ), then
  1. (i)

    the sequence {xk(t)}k∈ℕis convergent tox(t) for anyt∈[0,∞[,

     
  2. (ii)

    the sequence {xk}k∈ℕis convergent toxin\(L^{1}_{\mathrm{loc}}([0,\infty[, \mathbb{R})\).

     

The following theorem has been proved in [3, Theorem 7.1, p. 158]:

Theorem 3.4

A setBACloc([0,∞[,ℝ) is relatively weakly sequentially compact iff
  1. (i)

    the family\(C_{1}\vert_{T}= \{ \dot {x}\vert_{[0,T]}:x\in B \} \)is equiabsolutely summable2on [0,T] for anyT>0,

     
  2. (ii)

    the set {x(0):xB} is bounded in ℝ.

     

4 Lower Semicontinuity of an Integral Functional

Consider the integral functional
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equf_HTML.gif
where G:[0,T]×ℝn×ℝm→ℝ∪{+∞}.

The following theorem has been proved in [7].

Theorem 4.1

If
  1. (i)

    the functionGis a normal integrand3on [0,T]×(ℝn×ℝm),

     
  2. (ii)

    the functionG(t,x,⋅) is convex onmfor any (t,x)∈[0,T]×ℝn,

     
  3. (iii)
    there exist a constantM∈ℝ and a summable functionΨ:[0,T]→ℝ such that
    https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equg_HTML.gif
    for any (t,x,u)∈[0,T]×ℝn×ℝm,
     
then
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equh_HTML.gif
provided that the sequence {xk}k∈ℕconverges tox0inL1([0,T],ℝn) and the sequence {uk}k∈ℕconverges weakly tou0inL1([0,T],ℝm).

4.1 Case of a (Proper) Lebesgue Integral

Consider the integral functional
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equi_HTML.gif
where l:[0,∞[×ℝn×ℝm×ℝ→ℝ∪{+∞}.

Theorem 4.2

If
  1. (i)

    the functionlis a normal integrand on [0,∞[×(ℝn×ℝm+1),

     
  2. (ii)

    the functionl(t,x,⋅,⋅) is convex onm×ℝ for any (t,x)∈[0,∞[×ℝn,

     
  3. (iii)
    there exist a constantM∈ℝ and a summable functionΨ:[0,∞[ →ℝ such that
    https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equj_HTML.gif
    for (t,x,ξ,λ)∈[0,∞[×ℝn×ℝm×ℝ,
     
then
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equk_HTML.gif
provided that the sequence {xk}k∈ℕconverges tox0in\(L_{\mathrm{loc}}^{1}([0,\infty[,\mathbb{R}^{n})\), the sequence {ξk}k∈ℕconverges weakly toξ0in\(L_{\mathrm{loc}}^{1}([0,\infty[,\mathbb{R}^{m})\), the sequence {λk}k∈ℕL1([0,∞[,ℝ) converges weakly toλ0L1([0,∞[,ℝ) in\(L_{\mathrm{loc}}^{1}([0,\infty[,\mathbb{R})\), and
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equl_HTML.gif

Remark 4.1

If the functions x:[0,∞[ →ℝn, ξ:[0,∞[ →ℝm, and λ:[0,∞[ →ℝ are Lebesgue measurable and l is a normal integrand on [0,∞[×(ℝn×ℝm×ℝ), then the map [0,∞[∋tl(t,x(t),ξ(t),λ(t))∈ℝ is Lebesgue measurable. See [13, Corollary 2B].

Proof of Theorem 4.2

Notice that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equm_HTML.gif
The function l(⋅,xk(⋅),ξk(⋅),λk(⋅)) is, therefore, integrable on [0,∞[ as the sum of an integrable function (a nonnegative measurable function–see assumption (iii)) and a summable one. Consequently, \(I_{\smallint}(x_{k},\xi_{k},\lambda_{k})\) is well defined for k=0,1,… .

Step 1.Ψ≡0, M=0.

The function l(⋅,xk(⋅),ξk(⋅),λk(⋅)) is nonnegative in this case so
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ2_HTML.gif
(2)
for any T>0 and k=1,2,… .
The function \(l|_{[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{m}\times \mathbb{R}}\), treated as a function of (t,x,(ξ,λ)), satisfies the assumptions of Theorem 4.1. Moreover, the fact that the sequence {xk}k∈ℕ converges to x0 in \(L_{\mathrm{loc}}^{1}([0,\infty[,\mathbb{R}^{n})\) implies that the sequence {xk|[0,T]} converges to x0|[0,T] in L1([0,T],ℝn) for any T>0. The weak convergence of the sequence {ξk}k∈ℕ to ξ0 in \(L_{\mathrm{loc}}^{1}([0,\infty [,\mathbb{R}^{m})\) and the weak convergence of the sequence {λk}k∈ℕ to λ0 in \(L_{\mathrm{loc}}^{1}([0,\infty [,\mathbb{R})\) imply the weak convergence of the sequence {(ξk,λk)}k∈ℕ to (ξ0,λ0) in L1([0,T],ℝm+1) for any T>0. Hence, by Theorem 4.1,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ3_HTML.gif
(3)
for any T>0.
It follows from (2) and (3) that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equn_HTML.gif
The last equality results from the fact that l is nonnegative. Hence, we get the assertion for Ψ≡0 and M=0.

Step 2. The general case.

Consider the map
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equo_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equp_HTML.gif
The fact that Ψ is summable on [0,∞[ implies that the map (t,x,ξ,λ)→Ψ(t) is \(\mathcal{L}([0,\infty[)\times \mathcal{B}(\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R})\)-measurable. It can be inferred from the continuity of the map (t,x,ξ,λ)→ that it is \(\mathcal{L}([0,\infty[)\times \mathcal{B}(\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R})\)-measurable. For this reason, the map \(\mathcal{A}\) is \(\mathcal{L}([0,\infty [)\times\mathcal{B}(\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R})\)-measurable. Besides, \(\mathcal{A}(t,\cdot,\cdot,\cdot)\) is lower semicontinuous as the sum of the lower semicontinuous map l(t,⋅,⋅,⋅), the constant map Φ(t,⋅,⋅,⋅), and the continuous map ϒ(t,⋅,⋅,⋅). The map \(\mathcal{A}(t,x,\cdot,\cdot)\) is convex as the sum of the convex function l(t,x,⋅,⋅), the constant function Φ(t,x,⋅,⋅), and the linear map ϒ(t,x,⋅,⋅). Further, \(\mathcal{A}(t,x,\xi,\lambda)\geq0\) for (t,x,ξ,λ)∈[0,∞[×ℝn×ℝn×ℝ. Using the result obtained in Step 1, it can be deduced that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equq_HTML.gif
As a result,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equr_HTML.gif
The proof is over. □

4.2 Case of an Improper Lebesgue Integral

Now consider the functional
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equs_HTML.gif
where l:[0,∞[×ℝn×ℝm×ℝ→ℝ∪{+∞}.

Theorem 4.3

If
  1. (i)

    the functionlis a normal integrand on [0,∞[×(ℝn×ℝm+1),

     
  2. (ii)

    the functionl(t,x,⋅,⋅) is convex onm×ℝ for any (t,x)∈[0,∞[×ℝn,

     
  3. (iii)

    there exist a constantM∈ℝ and a locally summable functionΨ:[0,∞[ →ℝ that satisfy\(\lim_{T\rightarrow\infty}\int_{0}^{T}\varPsi(t)\,dt>-\infty\)andl(t,x,ξ,λ)≥Ψ(t)+for any (t,x,ξ,λ)∈[0,∞[×ℝn×ℝm×ℝ,

     
then
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equt_HTML.gif
provided that the sequence {xk}k∈ℕconverges tox0in\(L_{\mathrm{loc}}^{1}([0,\infty[,\mathbb{R}^{n})\); the sequence {ξk}k∈ℕconverges weakly toξ0in\(L_{\mathrm{loc}}^{1}([0,\infty[,\mathbb{R}^{m})\); \(\lambda_{k}\in L_{\mathrm{loc}}^{1}([0,\infty [,\mathbb{R})\); there exists a limit\(\lim_{T\rightarrow\infty}\int_{0}^{T}M\lambda_{k}(t)\,dt>-\infty\)fork=0,1,…; the sequence {λk}k∈ℕconverges weakly toλ0in\(L_{\mathrm{loc}}^{1}([0,\infty[,\mathbb{R})\); and
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equu_HTML.gif

Proof

It follows from the equality
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equv_HTML.gif
that the function l(⋅,xk(⋅),ξk(⋅),λk(⋅)) is locally integrable on [0,∞[ as the sum of an integrable function (a nonnegative measurable function) and a locally summable function. The existence of the limits
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equw_HTML.gif
implies the existence of the limit \(\lim_{T\rightarrow\infty}\int_{0}^{T}l(t,x_{k}(t),\xi_{k}(t),\lambda_{k}(t))\,dt\). Hence, Ilim(xk,ξk,λk) is well defined for k=0,1,… .

Step 1.Ψ≡0, M=0.

The function l(⋅,xk(⋅),ξk(⋅),λk(⋅)) is nonnegative in this case so
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ4_HTML.gif
(4)
for any S>0 and k=1,2,… .
Similarly as in the proof of Theorem 4.2,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ5_HTML.gif
(5)
for any T>0.
Using (4) and (5)
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equx_HTML.gif
This proves the assertion for Ψ≡0 and M=0.

Step 2. The general case.

Consider the map
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equy_HTML.gif
with
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equz_HTML.gif
The function \(\mathcal{A}\) satisfies the assumptions of Step 1, much in the same way as in Step 2 of the proof of Theorem 4.2. Hence,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equaa_HTML.gif
As a result,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equab_HTML.gif
The proof is completed. □

5 Modified Lagrangian

Let A:[0,∞[ ⇉ℝn be a multifunction with a closed graph \(\operatorname{Gr}A\) and let \({R}\colon\operatorname{Gr}A\rightrightarrows \mathbb{R}^{1+m}\) be a multifunction.

The multifunction R is said to have property (K) at a point \(( t,x_{0} ) \in\operatorname{Gr}A\) with respect to x iff
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equac_HTML.gif
By definition, the multifunction R has property (K) with respect to x iff it has property (K) at each point \(( t,x_{0} ) \in \operatorname{Gr}A\) with respect to x.

Remark 5.1

If R has property (K) with respect to x then it is obviously closed-valued.

We say that the multifunction R has property (Π) iff the fact that (η,ξ)∈R(t,x) implies that \(( \bar{\eta},\xi ) \in R(t,x)\) for \(\bar{\eta}\geq\eta\).

The modified Lagrangian4 is defined to be the function l:[0,∞[×ℝn×ℝm+1→ℝ∪{+∞} given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ6_HTML.gif
(6)
for any multifunction R. By agreement, inf∅=+∞.

Theorem 5.1

If a multifunctionRhas the\(\mathcal{L}([0,\infty [)\times\mathcal{B} ( \mathbb{R}^{n}\times\mathbb{R}^{m+1} ) \)-measurable graph and enjoys property (K) with respect toxand property (Π), then the modified Lagrangianlis a normal integrand on [0,∞[×(ℝn×ℝm+1). Moreover, ifRtakes convex values, then the functionl(t,x,⋅,⋅) is convex onm+1for any\((t,x)\in\operatorname{Gr}A\).

Proof

The \(\mathcal{L}([0,\infty[)\times\mathcal{B} ( \mathbb{R}^{n}\times\mathbb{R}^{m+1} ) \)-measurability of l can be proven using the same arguments as in the proof of the measurability of a Lagrangian defined on a bounded interval [0,T], presented in [5]. A proof of the lower semicontinuity of the function l(t,⋅,⋅,⋅) can be found in [14]. □

6 Lower Closure Theorems

We shall prove two lower closure theorems for functions defined on the interval [0,∞[.

6.1 Case of a (Proper) Lebesgue Integral

Theorem 6.1

Assume thatA:[0,∞[ ⇉ℝnis a multifunction with a closed graph\(\operatorname{Gr}A\)and\({R}\colon\operatorname{Gr}A \rightrightarrows \mathbb{R}^{1+m}\)is a convex-valued\(( \mathcal{L}([0,\infty[)\times\mathcal{B} ( \mathbb{R}^{n} ) ) \vert _{\mathrm{Gr}A}\)-measurable multifunction that has property (K) and property (Π). Letξk:[0,∞[→ℝm, xk:[0,∞[ →ℝn, ηk+1:[0,∞[ →ℝ, andλk:[0,∞[ →ℝ be measurable functions fork∈ℕ∪{0} such that
  1. (i)

    \(x_{k}(t)\in A(t)\ \mathit{for }t\in\lbrack0,\infty[\mathit{a.e.\ and\ each}\ k\in\mathbb{N}\),

     
  2. (ii)

    (ηk(t),ξk(t))∈R(t,xk(t)) for a.e. t∈[0,∞[ andeachk∈ℕ,

     
  3. (iii)
    the sequence {xk}k∈ℕconverges tox0in\(L_{\mathrm{loc}}^{1}([0,\infty[,\mathbb{R}^{n})\), the sequence {ξk}k∈ℕconverges weakly toξ0in\(L_{\mathrm{loc}}^{1}([0,\infty[,\mathbb{R}^{m})\), λkL1([0,∞[,ℝ) fork=0,1,… , the sequence {λk}k∈ℕconverges weakly toλ0in\(L_{\mathrm{loc}}^{1}([0,\infty[,\mathbb{R})\), and
    https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equae_HTML.gif
     
  4. (iv)

    ηk(t)≥λk(t) for a.e. t∈[0,∞[ andeachk∈ℕ,

     
  5. (v)

    \(\gamma:=\liminf_{k\rightarrow\infty}\int_{0}^{\infty}\eta_{k}(t)\,dt\in\mathbb{R}\).

     
Thenx0(t)∈A(t) for a.e. t∈[0,∞[ and there exists a summable functionη0:[0,∞[ →ℝ such that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equaf_HTML.gif

Remark 6.1

Since the multifunction R is \(( \mathcal{L}([0,\infty [)\times\mathcal{B} ( \mathbb{R}^{n} ) ) \vert_{\mathrm{Gr}A}\)-measurable, therefore, its graph is \(\mathcal{L}([0,\infty[)\times \mathcal{B} ( \mathbb{R}^{n} ) \times \mathcal{B} ( \mathbb{R}\times\mathbb{R}^{m} ) \)-measurable. This follows from [13, Theorem 1E, p. 164] and [9, Chap. I, Theorem 7.9, and Exercise 1, p. 325].

Proof of Theorem 6.1

The fact that x0(t)∈A(t) for a.e. t∈[0,∞[ follows immediately from assumptions (i) and (iii) and from the closedness of the set A(t). Let l:[0,∞[×ℝn×ℝm+1→ℝ∪{+∞} be the modified Lagrangian given by (6). By Theorem 5.1, l is normal and l(t,x,⋅,⋅) is convex on ℝm+1 for \((t,x)\in\operatorname{Gr}A\). Moreover, it follows from the definition of l that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ7_HTML.gif
(7)
for (t,x,ξ,λ)∈[0,∞[×ℝn×ℝm+1. Since the assumptions of Theorem 4.2 are satisfied, with Ψ≡0 and M=1, therefore,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equag_HTML.gif
It can be deduced from assumptions (ii), (iv), and (6) that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equah_HTML.gif
for a.e. t∈[0,∞[ and for each k∈ℕ. Hence, by (7),
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ8_HTML.gif
(8)
Further, the last inequality and the summability of the function λ0 on [0,∞[ imply that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ9_HTML.gif
(9)
Put
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equai_HTML.gif
for t∈[0,∞[. By assumption (v) and by (8),
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equaj_HTML.gif
In view of (9), this implies the summability of the function η0 on [0,∞[. Hence, η0 is finite a.e. on [0,∞[. As a result,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equak_HTML.gif
for a.e. t∈[0,∞[. Hence, for a.e. t∈[0,∞[ there exists a sequence {ηk}k∈ℕ, depending on t, such that (ηk,ξ0(t))∈R(t,x0(t)), ηkλ0(t), and limk→∞ηk=η0(t). By the closedness of the set R(t,x0(t)),
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equal_HTML.gif
for t∈[0,∞[ a.e. Obviously,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equam_HTML.gif
for t∈[0,∞[ a.e. Moreover, as was proven before,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equan_HTML.gif

The proof is completed. □

6.2 Case of an Improper Lebesgue Integral

Theorem 6.2

Assume thatA:[0,∞[ ⇉ℝnis a multifunction with a closed graph\(\operatorname{Gr}A\)and\({R}\colon\operatorname{Gr}A\rightrightarrows \mathbb{R}^{1+m}\)is\(( \mathcal{L}([0,\infty[)\times\mathcal{B} ( \mathbb{R}^{n} ) ) \vert_{\mathrm{Gr}A}\)-measurable, takes convex values, and has property (K) and property (Π). Letξk:[0,∞[ →ℝm, xk:[0,∞[ →ℝn, ηk+1:[0,∞[ →ℝ, andλk:[0,∞[ →ℝ be measurable functions fork∈ℕ∪{0} such that
  1. (i)

    xk(t)∈A(t) for a.e. t∈[0,∞[ and eachk∈ℕ,

     
  2. (ii)

    (ηk(t),ξk(t))∈R(t,xk(t)) for a.e. t∈[0,∞[ and eachk∈ℕ,

     
  3. (iii)
    the sequence {xk}k∈ℕconverges tox0in\(L_{\mathrm{loc}}^{1}([0,\infty[,\mathbb{R}^{n})\), the sequence {ξk}k∈ℕconverges weakly toξ0in\(L_{\mathrm{loc}}^{1}([0,\infty[,\mathbb{R}^{m})\), \(\lambda_{k}\in L_{\mathrm{loc}}^{1}([0,\infty [,\mathbb{R})\), there exists a limit\(\lim_{T\rightarrow\infty}\int_{0}^{T}\lambda_{k}(t)\,dt>-\infty\)fork=0,1,… , the sequence {λk}k∈ℕconverges weakly toλ0in\(L_{\mathrm{loc}}^{1}([0,\infty[,\mathbb{R})\), and
    https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equao_HTML.gif
     
  4. (iv)

    ηk(t)≥λk(t) for a.e. t∈[0,∞[ and eachk∈ℕ,

     
  5. (v)
    the functionsηkare locally summable and
    https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equap_HTML.gif
     
Thenx0(t)∈A(t) for a.e. t∈[0,∞[, and there exists a locally summable functionη0:[0,∞[ →ℝ such that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equaq_HTML.gif

Remark 6.2

Observe that the existence of the limit \(\lim_{T\rightarrow\infty}\int_{0}^{T}\eta_{k}(t)\,dt\) for k∈ℕ follows from assumption (iv), the local summability of λk, and the existence of the limit \(\lim_{T\rightarrow\infty}\int_{0}^{T}\lambda_{k}(t)\,dt\) for k∈ℕ.

Proof

The proof of Theorem 6.2 is based on Theorem 5.1. It is essentially the same as the proof of Theorem 6.1. The local summability of the function η0(t)=l(t,x0(t),ξ0(t),λ0(t)) on [0,∞[ and the existence of the limit \(\lim_{T\rightarrow\infty}\int_{0}^{T}\eta_{0}(t)\,dt\in\mathbb{R}\) follow from the inequalities
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equar_HTML.gif
and from the fact that the function λ0 is locally summable, and there exists a limit \(\lim_{T\rightarrow\infty}\int_{0}^{T}\lambda_{0}(t)\,dt>-\infty\). □

7 Existence of a Classical Optimal Pair

This section contains the main results of the paper.

Let us consider the infinite-horizon optimal control system (P) with the cost functional (\(J_{\smallint }\)). For this system, we introduce the definition of an admissible pair and of a classical optimal solution.

Assume that
(I1)

the multifunctions A:[0,∞[ ⇉ℝn and \(U\colon\operatorname{Gr}A\rightrightarrows \mathbb{R}^{m}\) have closed graphs \(\operatorname{Gr}A\) and \(\operatorname{Gr}U\);

(I2)

the set ⋃xZU(t,x) is bounded for each point t∈[0,∞[ and each bounded set Z⊂ℝn;

(I3)
f:[0,∞[×ℝn×ℝm→ℝn is a Carathéodory function5 with respect to t∈[0,∞[ and (x,u)∈ℝn×ℝm and satisfies the following growth condition: for any T>0 there exist a nonnegative summable function ΨT:[0,T]→ℝ and a constant A≥0 such that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equas_HTML.gif
for each \((t,x,u)\in\operatorname{Gr}U\);
(I4)

F:[0,∞[×ℝn×ℝm→ℝ is a Carathéodory function with respect to t∈[0,∞[ and (x,u)∈ℝn×ℝm;

(I5)
the multifunction \({Q}\colon\operatorname{Gr}A\rightrightarrows \mathbb{R}^{1+n}\) given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ10_HTML.gif
(10)
takes convex values.

The following elementary result holds true:

Theorem 7.1

If assumption (I1) is satisfied, then the multifunction\(U\colon\operatorname{Gr}A\rightrightarrows \mathbb{R}^{m}\)is\(( \mathcal{L}([0,\infty[)\times\mathcal{B} ( \mathbb{R}^{n} ) ) \vert_{\mathrm{Gr}A} \)-measurable.

Proof

Notice that if \(\operatorname{Gr}U\) is a closed set, then the set6U−1(C) is closed in \(\operatorname{Gr}A\) for every compact set C⊂ℝm; see [13, p. 165]. Next, consider an arbitrary compact set C⊂ℝm. Since the set U−1(C) is closed in [0,∞[×ℝn, therefore, it is \(\mathcal{B}([0,\infty[\times\mathbb{R}^{n})\)-measurable and consequently \(\mathcal{L}([0,\infty[)\times\mathcal{B}(\mathbb{R}^{n})\)-measurable. Hence, U−1(C) is an \(( \mathcal{L}([0,\infty[)\times\mathcal{B} ( \mathbb{R}^{n} ) ) \vert_{\mathrm{Gr}A}\)-measurable set. By [13, Proposition 1A, p. 160], the closedness of \(\operatorname{Gr}U\) implies the closedness of the values of the map U, which permits us to infer that the map U is \(( \mathcal{L}([0,\infty[)\times\mathcal{B} ( \mathbb{R}^{n} ) ) \vert_{\mathrm{Gr}A}\)-measurable. □

The measurability part of Theorem 7.2 has been proven in [5]. That proof is based on the Castaing representation theorem for multifunctions. The proof presented in this paper is based on properties of some special multifunctions. Property (K) of Q may be deduced from the fact that the modified Lagrangian is a normal integrand. We shall prove this property in a direct way.

Theorem 7.2

If assumptions (I1)(I5) are satisfied, then the multifunction\({Q}: \operatorname{Gr}A\rightrightarrows \mathbb{R}^{1+n}\)given by (10) is\((\mathcal{L}([0,\infty[)\times \mathcal{B} ( \mathbb{R}^{n})) \vert_{\mathrm{Gr}A}\)-measurable and has property (K) and property (Π).

Proof

Consider a multifunction \(\widetilde{Q}\colon\operatorname{Gr}A\rightrightarrows \mathbb{R}^{1+n+m}\) given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equat_HTML.gif
Next, let \(\widetilde{Q}_{1}\colon\operatorname{Gr}A\rightrightarrows \mathbb{R}^{1+n+m}\), \(\widetilde{Q}_{2}\colon\operatorname{Gr}A\rightrightarrows \mathbb{R}^{1+n+m}\), and \(\widetilde{Q}_{3}\colon\operatorname{Gr}A\rightrightarrows \mathbb{R}^{1+n+m}\) be given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equau_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equav_HTML.gif
It follows from the continuity of the function f1 that it is a Carathéodory function with respect to \((t,x)\in\operatorname{Gr}A\) and (η,ξ,u)∈ℝ1+n+m, i.e., the function f1(⋅,⋅,η,ξ,u) is measurable with respect to the σ-algebra \(( \mathcal{L}([0,\infty[)\times\mathcal{B}(\mathbb{R}^{n}) ) |_{\mathrm{Gr}A}\) for each (η,ξ,u)∈ℝ1+n+m, and the function f1(t,x,⋅,⋅,⋅) is continuous on ℝ1+n+m for each \((t,x)\in\operatorname{Gr}A\). It can be inferred from Theorem 7.1 that the map \(U\colon\operatorname{Gr}A\rightrightarrows \mathbb{R}^{m}\) is measurable. Since the graph \(\operatorname{Gr}U\) is closed, the values of U are closed. Hence, by [13, Corollary 1Q], the map \(\widetilde{Q}_{1}\) is \(( \mathcal{L}([0,\infty[)\times \mathcal{B} ( \mathbb{R}^{n} ) ) \vert_{\mathrm{Gr}A}\)-measurable and takes closed values.
Consider the map
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equaw_HTML.gif
It can be deduced from assumption (I4) and [13, Proposition 2A and Proposition 2C] that this map is a Carathéodory function with respect to \((t,x)\in\operatorname{Gr}A\) and (η,ξ,u)∈ℝ×ℝn×ℝm—and so is the map f2, as a consequence. It means that the map \(\widetilde{Q}_{2}\) is \(( \mathcal{L}([0,\infty[)\times\mathcal{B} ( \mathbb{R}^{n} ) ) \vert_{\mathrm{Gr}A}\)-measurable and closed-valued by [13, Theorem 2I]; the fact that f2 is a normal integrand follows from [13, Theorem 2C]. Similarly, the map f3 is a Carathéodory function with respect to \((t,x)\in\operatorname{Gr}A\) and (η,ξ,u)∈ℝ×ℝn×ℝm.
Consider, for i=1,…,n, the maps
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equax_HTML.gif
given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equay_HTML.gif
where \(f_{3}^{i}\) is the ith coordinate function of f3. It follows from [13, Theorem 2I] that the maps \(\widetilde {Q}_{3}^{i}\) and \(\widetilde{P}_{3}^{i}\) are \((\mathcal{L}([0,\infty[)\times\mathcal{B} ( \mathbb{R}^{n} ) ) \vert_{\mathrm{Gr}A}\)-measurable and closed-valued. Thus, the map
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equaz_HTML.gif
is \(( \mathcal{L}([0,\infty[)\times\mathcal{B} ( \mathbb{R}^{n} ) ) \vert_{\mathrm{Gr}A}\)-measurable and closed-valued by [13, Corollary 1M]. Obviously,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equba_HTML.gif
for \((t,x)\in\operatorname{Gr}(A)\). Consequently, [13, Corollary 1M] implies that the map \(\widetilde{Q}\) is \(( \mathcal{L}([0,\infty[)\times\mathcal{B} ( \mathbb{R}^{n} ) ) \vert_{\mathrm{Gr}A}\)-measurable and closed-valued.
Now choose any \((t,x)\in\operatorname{Gr}A\) and consider the continuous map G(t,x):ℝ×ℝn+m→ℝ×ℝn given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbb_HTML.gif
It follows from the continuity of G(t,x) that its graph is closed. As a result, the graph of the multifunction
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbc_HTML.gif
is closed. Further, the constant multifunction
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbd_HTML.gif
is closed-valued and \(( \mathcal{L}([0,\infty[)\times\mathcal{B} ( \mathbb{R}^{n} ) ) \vert_{\mathrm{Gr}A}\)-measurable; see [13, Proposition 1A]. It can be inferred directly from the definitions that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Eqube_HTML.gif
for \((t,x)\in\operatorname{Gr}A\). Since the set U(t,x) is compact (see assumptions (I1) and (I2)), therefore, the set \(\widetilde{G}_{(t,x)} ( \widetilde{Q}(t,x) ) \) is closed, i.e.,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbf_HTML.gif
for \((t,x)\in\operatorname{Gr}A\). By [13, Theorem 1N], applied to the maps \(\widetilde{Q}\) and \(\widetilde{G}_{(t,x)}\), the map Q is \(( \mathcal{L}([0,\infty[)\times\mathcal{B} ( \mathbb{R}^{n} ) ) \vert_{\mathrm{Gr}A}\)-measurable and closed-valued.
We shall show that the multifunction Q has property (K). Indeed, consider an arbitrary point \((t,x_{0})\in\operatorname{Gr}A\). Obviously,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbg_HTML.gif
To prove the reverse inclusion, consider an arbitrary point (η,ξ)∈ℝ×ℝn such that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbh_HTML.gif
Then
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbi_HTML.gif
for any δ>0. Hence, for any k∈ℕ there exists a point
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbj_HTML.gif
such that \(| ( \eta,\xi ) - ( \eta_{k},\xi_{k} ) |<\frac{1}{k}\). As a result,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbk_HTML.gif
and, for any k∈ℕ, there exists xk such that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbl_HTML.gif
Hence, the sequence {xk}k∈ℕ converges to x0. It follows from the closedness of \(\operatorname{Gr}A\) that x0A(t). Next, by the definition of Q there exists a sequence {uk}k∈ℕ such that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbm_HTML.gif
for any k∈ℕ. It can be inferred from the boundedness of the sequence {xk}k∈ℕ and from assumption (I2) that there exists a subsequence of the sequence {uk}k∈ℕ, still denoted by {uk}k∈ℕ, that converges to some u0. It follows from the closedness of the graph \(\operatorname{Gr}U\) and the convergence of the sequence {xk}k∈ℕ that u0U(t,x0). The continuity of the function F(t,⋅,⋅) implies
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbn_HTML.gif
The continuity of the function f(t,⋅,⋅) leads to the conclusion that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbo_HTML.gif
Thus, (η,ξ)∈Q(t,x0), which means that Q has property (K).

The fact that Q has property (Π) follows immediately from the definition of Q. □

Using Gronwall’s lemma, one can prove the following.

Theorem 7.3

If a functionf:[0,∞[×ℝn×ℝm→ℝnsatisfies assumption (I3), then the set of all admissible trajectories is a relatively sequentially weakly compact subset ofACloc([0,∞[,ℝn) (this set may be empty).

Remark 7.1

A similar theorem has been proven in [3] under a stronger assumption on f, the so-called growth condition (γ). The set of all admissible trajectories is nonempty under that assumption; see [6]. By applying an analogous method, we can obtain the compactness of the set of admissible trajectories under the weaker assumption (I3). However, (I3) does not ensure that the set of all admissible trajectories is nonempty.

7.1 Existence of a Classical Optimal Solution

In this section, we can state and prove a theorem on the existence of a classical optimal solution to the problem described by (P) and (\(J_{\smallint }\)).

Definition 7.4

A pair of functions (x,u):[0,∞[ →ℝn×ℝm is called admissible for the optimal control problem given by (P) and (\(J_{\smallint }\)) if xACloc([0,∞[,ℝn), u is measurable, the pair (x,u) satisfies system (P), and the function F(⋅,x(⋅),u(⋅)) is integrable on [0,∞[ (not necessarily summable).

The set of all admissible pairs (x,u) introduced in Definition 7.4 is denoted by \(\varOmega_{\smallint}\). A function xACloc([0,∞[,ℝn) is called an admissible trajectory if there exists a measurable function u such that \((x,u)\in\varOmega_{\smallint}\).

Definition 7.5

A pair \((x^{\ast},u^{\ast})\in\varOmega_{\smallint}\) is called a classical optimal solution to the problem given by (P) and (\(J_{\smallint }\)) if \(J_{\smallint}(x^{\ast},u^{\ast})\in\mathbb{R}\) and
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbp_HTML.gif
for each pair \((x,u)\in\varOmega_{\smallint}\).
Assume that:
\((I_{\smallint}6)\)
there exists a constant α∈ℝ such that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbq_HTML.gif
\((I_{\smallint}7)\)
there exists a summable function λ:[0,∞[ →ℝ such that for each pair \((x,u)\in\varOmega_{\smallint}^{\alpha}\)
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbr_HTML.gif

Theorem 7.6

If assumptions (I1)(I5), \((I_{\smallint}6)\), and\((I_{\smallint}7)\)are satisfied, then the problem given by (P) and (\(J_{\smallint }\)) has a classical optimal solution.

Proof

Assumption \((I_{\smallint}6)\) implies that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ11_HTML.gif
(11)
Consequently, by \((I_{\smallint}7)\)
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ12_HTML.gif
(12)
Let \(\{(x_{k},u_{k})\}_{k\in\mathbb{N}}\subset\varOmega_{\smallint}^{\alpha}\) be such a sequence that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ13_HTML.gif
(13)
By Theorem 7.3, one can choose a subsequence of the sequence {xk}k∈ℕ, that converges weakly in ACloc([0,∞[,ℝn) to some xACloc([0,∞[,ℝn). Without loss of generality, this subsequence shall be denoted by {xk}k∈ℕ. Define
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbs_HTML.gif
Since \(\{(x_{k},u_{k})\}_{k\in\mathbb{N}}\subset\varOmega_{\smallint}^{\alpha}\), the functions ηk are summable on [0,∞[ for k∈ℕ and \(\liminf_{k\rightarrow\infty}\int_{0}^{\infty}\eta_{k}(t)\,dt\in\mathbb{R}\). By Theorem 3.3(i), the sequence {xk(t)}k∈ℕ converges to x(t) for any t∈[0,∞[. Using the closedness of the graph \(\operatorname{Gr}A\) and the fact that xk(t)∈A(t) for t∈[0,∞[, k∈ℕ, it can be deduced that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbt_HTML.gif
In view of Theorem 3.2(i), the sequence \(\{ \dot{x}_{k}\vert_{D_{T}} \}_{k\in\mathbb{N}}\) converges weakly to \(\dot{x}\vert_{D_{T}}\) in L1([0,T],ℝn) for any T>0. The multifunction Q satisfies the assumptions of Theorem 6.1 that concern R (this follows from Theorem 7.2 and assumption (I5)) and the sequence {(ηk,ξk)}k∈ℕ satisfies condition (ii) of Theorem 6.1. The constant sequence {λk}k∈ℕ converges weakly to λ0=λ in \(L_{\mathrm{loc}}^{1}([0,\infty[,\mathbb{R})\) in an obvious way and
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbu_HTML.gif
By \((I_{\smallint}7)\), ηk(t)≥λk(t) for a.e. t∈[0,∞[ and k∈ℕ. Applying Theorem 3.3(ii) leads to the conclusion that the sequence {xk}k∈ℕ converges to x0 in \(L_{\mathrm{loc}}^{1}([0,\infty[,\mathbb{R})\). Thus, the assumptions of Theorem 6.1 are satisfied. Consequently, there exists a summable function η0:[0,∞[ →ℝ such that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ14_HTML.gif
(14)
Now consider the multifunction Γ:[0,∞[ ⇉ℝm given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbv_HTML.gif
Note that the closed-valued multifunction [0,∞[∋tU(t,x(t))∈ℝm is measurable; the closedness of the values follows from the closedness of the graph \(\operatorname{Gr}U\). Indeed, consider the closed-valued multifunction \(\overline{\varGamma}\colon \lbrack0,\infty[\ \rightrightarrows \mathbb{R}^{1+n}\) given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbw_HTML.gif
The measurability of this multifunction follows from the continuity of the function [0,∞[∋t→(t,x(t))∈ℝ×ℝn. Next, for any fixed t∈[0,∞[, consider the multifunction At:ℝ×ℝn⇉ℝm given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbx_HTML.gif
It is easy to see that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equby_HTML.gif
Hence, \(\operatorname{Gr}A_{t}\) is closed. Moreover, the multifunction
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equbz_HTML.gif
is measurable as a constant closed-valued map; see [13, Proposition 1A]. By [13, Theorem 1N], the multifunction
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equca_HTML.gif
is measurable. In view of [13, Theorem 2N and Proposition 2C], the map G :[0,∞[×ℝm→ℝ given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcb_HTML.gif
is a normal integrand and the map H:[0,∞[×ℝm→ℝn given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcc_HTML.gif
is a Carathéodory function. Since (η0(t),ξ0(t))∈Q(t,x(t)) for a.e. t∈[0,∞[, therefore, the map Γ has nonempty values a.e. on [0,∞[. Using the implicit function theorem for multifunctions ([13, Theorem 2J]), notice that the map Γ is measurable, closed-valued, and there exists a measurable function u:[0,∞[ →ℝm such that u(t)∈Γ(t) for a.e. t∈[0,∞[. It follows from the description of Γ that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcd_HTML.gif
The pointwise convergence of admissible trajectories xk(t) to x(t) on [0,∞[ and the initial conditions xk(0)=0 for k∈ℕ imply that x(0)=0. As a result, the pair (x,u) satisfies (P). Moreover, it follows from the definition of Γ that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ15_HTML.gif
(15)
Thus, the summability of the function η0 implies the integrability of the function
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equce_HTML.gif
Finally, \((x^{\ast},u^{\ast})\in\varOmega_{\smallint}\). Apply this fact and (11), (13), (14), and (15) to obtain
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcf_HTML.gif
Hence, by (12),
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcg_HTML.gif
and the proof is completed. □

Example 7.1

Consider the control system given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Figd_HTML.gif
where a∈ℝ∖{0}, b∈ℝ∖{0}, x∈ℝ, u∈ℝ, A(t)=ℝ, \(U(t,x)= [ -\frac{1}{t^{2}+1},\frac{1}{t^{2}+1} ] \) for t∈[0,∞[, x∈ℝ. This is a special case of system (P).

It is easy to see that there exists a unique solution xACloc([0,∞[,ℝ) of system (P1), that corresponds to any fixed measurable control u:[0,∞[ →ℝ such that \(u(t)\in [ -\frac{1}{t^{2}+1},\frac{1}{t^{2}+1} ] \) for a.e. t∈[0,∞[.

Assume that the cost functional is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Fige_HTML.gif
It is easy to see that the integral \(\int_{0}^{\infty} ( \frac{\sin^{2} ( x(t) ) u^{2}(t)}{1+t^{2}}+u(t) ) \,dt\) exists and is finite for an arbitrary pair (x,u) such that xACloc([0,∞[,ℝ) and u:[0,∞[ →ℝ is a measurable function that satisfies system (P1). Hence, \(\varOmega_{\smallint}\neq\emptyset\).
Moreover, the optimal control problem given by (P1) and (\(J_{\smallint }^{1}\)) satisfies the assumptions of Theorem 7.6 with \(F(t,x,u)=\frac{\sin^{2}(x)u^{2}}{1+t^{2}}+u\). Indeed, assumptions (I1)–(I4) are fulfilled in an obvious way. From the convexity of the set U(t,x) and from the convexity of the function,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equch_HTML.gif
it follows that the set Q(t,x) is convex, i.e., assumption (I5) is satisfied.
Moreover, for \(\widetilde{u}(t)=0\)
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equci_HTML.gif
\(\widetilde{x}=0\) is the unique solution of system (P1), corresponding to \(\widetilde{u}=0\). Hence, \(\varOmega_{\smallint}^{0}\neq\emptyset\), i.e. assumption \((I_{\smallint}6)\) is satisfied with α=0. Finally,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcj_HTML.gif
for any \((x,u)\in\varOmega_{\smallint}^{0}\). Thus, assumption \((I_{\smallint}7)\) is satisfied with \(\lambda(t)=-\frac{1}{1+t^{2}}\). The function \([0,\infty[\ \ni t\rightarrow-\frac{1}{1+t^{2}}\in\mathbb{R}\) is obviously summable. Consequently, the control system given by (P1) and (\(J_{\smallint }^{1}\)) has a classical optimal solution (x,u).

7.2 Existence of an Almost Strongly Optimal Solution

First, we define an admissible pair and an almost strongly optimal pair for the problem (P), with the functional given by (\(J_{\qopname \relax m{lim}}\)).

Definition 7.7

A pair of functions (x,u):[0,∞[ →ℝn×ℝm is called admissible for the optimal control system (P) with the functional Jlim iff xACloc([0,∞[,ℝn), u is measurable function, the pair (x,u) satisfies system (P), a function F(t,x(t),u(t)) is locally integrable on [0,∞[, and there exists a limit
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equck_HTML.gif
(not necessarily finite).

The set of all admissible pairs (x,u), in the sense of Definition 7.7, will be denoted by Ωlim. A function xACloc([0,∞[,ℝn) is called an admissible trajectory iff there exists a measurable function u such that (x,u)∈Ωlim.

Definition 7.8

A pair (x,u)∈Ωlim is called almost strongly optimal iff Jlim(x,u)∈ℝ and
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcl_HTML.gif
for any pair (x,u)∈Ωlim.
We require that the following conditions hold true:
(Ilim6)
there exists a constant α∈ℝ such that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcm_HTML.gif
(Ilim7)
there exists a locally summable function λ:[0,∞[ →ℝ such that \(\lim _{T\rightarrow\infty}\int_{0}^{T}\lambda ( t ) \,dt>-\infty\) and, for any pair \((x,u)\in\varOmega_{\lim}^{\alpha}\),
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcn_HTML.gif

We have the following theorem on the existence of an almost strongly optimal solution.

Theorem 7.9

If assumptions (I1)(I5), (Ilim6), and (Ilim7) are satisfied, then the problem given by (P) and (\(J_{\qopname \relax m{lim}}\)) has an almost strongly optimal solution.

Proof

The fact that \(\varOmega_{\lim}^{\alpha}\neq\emptyset\) for some α∈ℝ (see Ilim6) implies that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ16_HTML.gif
(16)
By assumption (Ilim7),
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ17_HTML.gif
(17)
Let \(\{(x_{k},u_{k})\}_{k\in\mathbb{N}}\subset\varOmega_{\lim}^{\alpha}\) be a sequence such that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ18_HTML.gif
(18)
By Theorem 7.3, one can choose a subsequence of the sequence {xk}k∈ℕ, weakly convergent in ACloc([0,∞[,ℝn) to some xACloc([0,∞[,ℝn). Without loss of generality, we shall denote this subsequence by {xk}k∈ℕ. Define the functions ηk, x0, ζk, and λk as in the proof of Theorem 7.6. Since \(\{(x_{k},u_{k})\}_{k\in\mathbb{N}}\subset\varOmega_{\lim}^{\alpha}\), therefore the functions ηk are locally summable on [0,∞[ for k∈ℕ. It can be checked in the same way as in the proof of Theorem 7.6 that the sequence {xk(t)}k∈ℕ converges to x(t) for any t∈[0,∞[,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ19_HTML.gif
(19)
and the assumptions of Theorem 6.2 are satisfied with R=Q (see (I5)); the fact that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equco_HTML.gif
follows from (17) and (18). As a result, there exists a locally summable function η0:[0,∞[ →ℝ such that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ20_HTML.gif
(20)
Consider now the multifunction Γ:[0,∞[ ⇉ℝm given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcp_HTML.gif
It follows from the implicit function theorem for multifunctions [13, Theorem 2J], that there exists a measurable function u:[0,∞[ →ℝm such that u(t)∈Γ(t) for a.e. t∈[0,∞[; see the proof of Theorem 7.6 for details. By the description of Γ,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ21_HTML.gif
(21)
The pointwise convergence of the trajectories xk(t) to x(t) on [0,∞[ and the conditions xk(0)=0, k∈ℕ, lead to the conclusion that x(0)=0. Thus, the pair (x,u) satisfies system (P); see (19). Since η0 is locally summable on [0,∞[, (21) implies that the function F(t,x(t),u(t)) is locally integrable on [0,∞[. Using the fact that \(\lim_{T\rightarrow\infty}\int_{0}^{T}\eta_{0}(t)\,dt\in\mathbb{R}\) and that there exists a limit
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcq_HTML.gif
as a limit of a nonincreasing function (see (21)), we can claim that there exists a limit
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcr_HTML.gif
finite or equal to −∞. Thus, (x,u)∈Ωlim. From (16), (18), (20), and (21)
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcs_HTML.gif
This means that the pair (x,u) is almost strongly optimal, since by (17)
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equct_HTML.gif
The proof is completed. □

Example 7.2

Consider a problem
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Figf_HTML.gif
where a∈ℝ∖{0}, b∈ℝ∖{0}, x∈ℝ, u∈ℝ, A(t)=ℝ, and
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcu_HTML.gif
for t∈[0,∞[ and x∈ℝ. System (P2) is a special case of system (P).
Consider the cost functional given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Figg_HTML.gif
System (P2), with the functional (\(J_{\qopname \relax m{lim}}^{2}\)), satisfies assumptions (I1)–(I5). Consider the control \(\widetilde{u}\colon \lbrack0,\infty[\ \rightarrow\mathbb{R}\) given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcv_HTML.gif
The control \(\widetilde{u}\) and the corresponding trajectory \(\widetilde{x}\in AC_{\mathrm{loc}}([0,\infty[,\mathbb{R})\) satisfy system (P2), the function \(F ( t,\widetilde{x}(t),\widetilde{u}(t) ) \) is locally summable on [0,∞[, and there exists
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcw_HTML.gif
This means that \((\widetilde{x},\widetilde{u})\in\varOmega_{\lim}\).
Moreover,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcx_HTML.gif
Therefore, \(\varOmega_{\lim}^{\frac{\pi}{2}}\neq\emptyset\) and assumption (Ilim6) is satisfied. Finally, observe that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcy_HTML.gif
for any pair \((x,u)\in\varOmega_{\lim}\supset\varOmega_{\lim}^{\frac{\pi}{2}}\). This means that assumption (Ilim7) is satisfied for \(\lambda(t)=\widetilde {u}(t)\) (the function \(\widetilde{u}\) is locally summable on [0,∞[ and \(\lim_{T\rightarrow\infty}\int_{0}^{T}\widetilde{u}(t)\,dt=0\)). Theorem 7.9 implies that the control system (P2) with the cost functional (\(J_{\qopname \relax m{lim}}^{2}\)) has an almost strongly optimal solution (x,u).

Observe that the pair \((\widetilde{x},\widetilde{u})\) is not admissible for the control problem (P1) with the cost functional (\(J_{\smallint }^{1}\)).

8 Optimality Principles

Let us introduce the following definition; see [3].

Definition 8.1

A pair (x,u) where xACloc([0,∞[,ℝn) and u:[0,∞[ →ℝm is a measurable function is called finitely optimal iff it satisfies system (P) on [0,T]; the integral \(\int_{0}^{T}F ( t,x^{\ast} ( t ) ,u^{\ast} ( t ) ) \,dt\) is finite; and the inequality
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equcz_HTML.gif
holds true for each pair (x,u):[0,T]→ℝn×ℝm where x is an absolutely continuous function on [0,T] and u is a measurable function on [0,T], such that they satisfy system (P) on [0,T] and the condition
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ22_HTML.gif
(22)
and such that the function F(⋅,x(⋅),u(⋅)) is integrable on [0,T].

8.1 Case of a (Proper) Lebesgue Integral

The method of proving the next theorem is similar to that presented in [3, Theorem 2.2]. We give this proof here to complete the task.

Theorem 8.2

If a pair\((x^{\ast},u^{\ast})\in \varOmega_{\smallint}\)is a classical optimal pair then it is finitely optimal.

Proof

Assume, for contradiction, that a classical optimal pair \((x^{\ast},u^{\ast})\in\varOmega_{\smallint}\) is not finitely optimal. Then there exist some T>0 and a pair of functions (x+,u+):[0,T]→ℝn×ℝm where x+ is absolutely continuous on [0,T] and u+ is measurable on [0,T] such that they satisfy system (P) on [0,T], condition (22), and the inequality
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equda_HTML.gif
In this case, there exists an ε>0 such that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ23_HTML.gif
(23)
Let \((\tilde{x},\tilde{u})\colon\lbrack0,\infty[\ \rightarrow\mathbb{R}^{n}\times\mathbb{R}^{m}\) be defined by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ24_HTML.gif
(24)
First, we shall prove that \((\tilde{x},\tilde{u})\in\varOmega_{\smallint}\); see Definition 7.4. Obviously, \(\tilde{x}\in AC_{\mathrm{loc}}([0,\infty[,\mathbb{R}^{n})\), the function \(\tilde{u}\) is measurable, and the pair \((\tilde{x},\tilde{u})\) satisfies system (P) on [0,∞[. Moreover, the function \(F ( \cdot,\tilde{x}(\cdot),\tilde{u}(\cdot) ) \) is integrable on [0,∞[. Indeed, since the integral \(\int_{0}^{T}F ( t,x^{+} ( t ) ,u^{+} ( t ) ) \,dt\) exists, therefore,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equdb_HTML.gif
where [F(⋅,x+(⋅),u+(⋅))]+ and [F(⋅,x+(⋅),u+(⋅))] are the positive and negative parts of the function F(⋅,x+(⋅),u+(⋅)), and at least one of the integrals on the right-hand side is finite; see the definition of integrability in Sect. 2. Assume that the integral
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equdc_HTML.gif
is finite. The pair (x,u) is a classical optimal pair, hence
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equdd_HTML.gif
and both integrals on the right-hand side are finite, because \(J_{\smallint }(z^{\ast},u^{\ast})\in\mathbb{R}\). Since
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equde_HTML.gif
therefore,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equdf_HTML.gif
the second integral on the right-hand side is finite because the integral \(\int_{0}^{\infty} [ F ( t,x^{\ast} ( t ) ,u^{\ast} ( t ) ) ]_{+}\,dt\) is finite. As a result, the integral \(\int_{0}^{\infty}F ( t,\widetilde{x} ( t ) , \widetilde{u} ( t ) ) \,dt\) exists. In a similar way, one can consider the case where the integral \(\int_{0}^{T} [ F ( t,x^{+} ( t ) ,u^{+} ( t ) ) ]_{-}\,dt\) is finite. Hence, \((\widetilde{x},\widetilde{u})\in\varOmega_{\smallint}\). Since the pair (x,u) is a classical optimal pair and (23) holds true, therefore,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equdg_HTML.gif
The contradiction completes the proof. □

8.2 Case of an Improper Lebesgue Integral

Theorem 8.3

If a pair (z,u)∈Ωlimis almost strongly optimal, then it is finitely optimal.

Proof

(7) Assume that the statement does not hold. Then there exists some T>0 and some pair of functions (x+,u+) : [0,T] → ℝn×ℝm where x+ is absolutely continuous on [0,T] and u+ is measurable on [0,T], such that they satisfy system (P) on [0,T], the condition x(T)=x+(T), and the inequality
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equdh_HTML.gif
Let ε>0 be such that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ25_HTML.gif
(25)
Consider the pair \((\tilde{x},\tilde{u})\colon\lbrack0,\infty[\ \rightarrow \mathbb{R}^{n}\times\mathbb{R}^{m}\) given by (24). First, we shall prove that \((\tilde{x},\tilde{u})\in\varOmega_{\lim}\). Obviously, \(\tilde{x}\in AC_{\mathrm{loc}}([0,\infty[,\mathbb{R}^{n})\), \(\tilde{u}\) is a measurable function on [0,∞[, and \((\widetilde{x},\widetilde{u})\) satisfies system (P) on [0,∞[. Moreover, the function \(F ( \cdot,\tilde{x}(\cdot),\tilde{u}(\cdot) ) \) is locally integrable on [0,∞[ and there exists a limit
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equdi_HTML.gif
not necessarily finite. Indeed, since the integral \(\int_{0}^{T}F ( t,x^{+} ( t ) ,u^{+} ( t ) ) \,dt\) exists, the following equality holds true:
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equdj_HTML.gif
where [F(⋅,x+(⋅),u+(⋅))]+ and [F(⋅,x+(⋅),u+(⋅))] are the positive and negative parts of the function F(⋅,x+(⋅),u+(⋅)) and at least one of the integrals on the right-hand side is finite. Assume that the integral \(\int_{0}^{T} [ F ( t,x^{+} ( t ) ,u^{+} ( t ) ) ]_{+}\,dt\) is finite. Since the pair (x,u) is almost strongly optimal, therefore,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equdk_HTML.gif
for any T1>0, and in this case the integrals on the right-hand side are finite. Consequently, for any T1>T
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equdl_HTML.gif
This means that the integral \(\int_{0}^{T_{1}} [ F ( t,\widetilde {x} ( t ) ,\widetilde{u} ( t ) ) ]_{+}\,dt\) is finite and, as a result, the integral \(\int_{0}^{T_{1}}F ( t,\widetilde{x} ( t ) ,\widetilde{u} ( t ) ) \,dt\) exists for any T1>0; the existence of such an integral for T1T follows from the fact that \((\tilde{x},\tilde{u})|_{[0,T]}=(x^{+},u^{+})\). We shall prove that the limit
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equdm_HTML.gif
exists, not necessarily finite. Indeed, for T1>T
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equ26_HTML.gif
(26)
Since the pair (x,u) is almost strongly optimal, there exists a finite limit
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equdn_HTML.gif
(the integral \(\int_{0}^{T}F ( t,x^{\ast} ( t ) ,u^{\ast} ( t ) ) \,dt\) is finite, too) and, further, there exists a limit
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equdo_HTML.gif
Hence, \((\tilde{x},\tilde{u})\in\varOmega_{\lim}\). By the fact that the pair (x,u) is almost strongly optimal, by (25), and (26),
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0126-2/MediaObjects/10957_2012_126_Equdp_HTML.gif
The contradiction completes the proof. □

9 Concluding Remarks

We have considered an infinite-horizon optimal control problem with a cost functional given either by an integral over an unbounded interval (a Lebesgue integral) or by a limit of integrals (an improper Lebesgue integral). We have proposed natural definitions of optimality for these two models and stated some sufficient conditions for the existence of optimal solutions. The existence theorems are proven using the modified Lagrangian and some extensions of the lower closure theorem. The new definitions are compatible with the definitions for finite-horizon models (Theorem 8.2 and Theorem 8.3).

It seems reasonable to assume that similar tools can be used to determine sufficient conditions for the existence of optimal pairs for some models with cost functionals described by the lower and upper limits of Lebesgue integrals.

Footnotes
1

We assume that all multifunctions in this paper have nonempty sets as values.

 
2

Let E be a Lebesgue measurable and bounded subset of ℝ. A family of summable functions {fs:E→ℝ; sS}, where S is an arbitrary nonempty set of indices, is equiabsolutely summable on E iff for any ε>0 there exists a δ>0 such that ∫F|fs|≤ε for any sS and for any measurable set FE with |F|<δ, where |F| is the Lebesgue measure of F.

 
3

A function f:[0,T]×ℝk→ℝ∪{±∞} is a normal integrand iff it is \(\mathcal{L}([0,T])\times\mathcal{B}(\mathbb{R}^{k})\)-measurable and the function f(t,⋅) is lower semicontinuous on ℝk for any t∈[0,T]. Here, \(\mathcal{L}([0,T])\) is the family of Lebesgue measurable subsets of the interval [0,T] and \(\mathcal{B}(\mathbb{R}^{k})\) is the family of Borel subsets of ℝk.

 
4
The modified Lagrangian for optimal control problems was introduced by Erik J. Balder in 1982. The classical Lagrangian corresponds to λ=−∞ and was used in optimal control problems (see [4]) in connection with the deparameterization procedure. Another similar idea is the Lagrangian auxiliary function which is defined for (t,x,ξ)∈GrA×ℝm
$$\mathcal{L}(t,x,\xi):=\mathit{dist}\bigl(\xi, R(t,x)\bigr)$$
where dist denotes the Euclidean distance; the details can be found in [5]. It is worth mentioning that there is a close relationship between an auxiliary function and a separation function, which has been considered in [15, Chap. 5].
 
5

A function f:[0,∞[×ℝn×ℝm→ℝn is said to be a Carathéodory function iff f(⋅,x,u) is measurable for any (x,u)∈ℝn×ℝm and f(t,⋅,⋅) is continuous for any t∈[0,∞[.

 
6

\(U^{-1}(C):=\{(t,x)\in\operatorname{Gr}A\colon\ U(t,x)\cap C\neq\emptyset\}\).

 
7

The proof of Theorem 8.3 is much the same as the proof of Theorem 8.2.

 

Acknowledgements

The author is very grateful to Prof. Dariusz Idczak of the University of Łódź for his advice on this paper. The author would like to thank Dr. Marian Jakszto of the University of Łódź for pointing out the discussion of the improper Lebesgue integral in the literature.

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Authors and Affiliations

  1. 1.Department of Econometrics, Faculty of Economics and SociologyUniversity of ŁódźŁódźPoland