Abstract
We consider a first-order periodic system involving a time-dependent maximal monotone map, a subdifferential term, and a multivalued perturbation F(t, x). We prove existence theorems for the “convex” problem (that is, F is convex valued and for the “nonconvex” problem (that is, F is nonconvex valued). Also, we establish the existence of extremal trajectories (that is, solutions when the multivalued perturbation F(t, x) is replaced by ext F(t, x), the extreme points of F(t, x)). Also, we show that every solution of the convex problem can be approximated uniformly by certain extremal trajectories (“strong relaxation” theorem). Finally, we illustrate our result by examining a nonlinear periodic feedback control system.
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1 Introduction
In the present work, we study the following nonlinear multivalued periodic system:
In this problem, \(A\colon T\times \mathbb {R}^{N}\longrightarrow 2^{\mathbb {R}^{N}}\setminus \emptyset \) is a multivalued map which is maximal monotone in \(x\in \mathbb {R}^{N}\), \(\varphi \in {\Gamma }_{0}(\mathbb {R}^{N})\) (the cone of lower semicontinuous, convex, proper functions; see Section 2) with ∂φ being the subdifferential in the sense of convex analysis and \(F\colon T\times \mathbb {R}^{N}\longrightarrow 2^{\mathbb {R}^{N}}\setminus \emptyset \) is a multivalued perturbation. We prove existence theorems for problem (1.1) when F is convex valued (“convex problem”) and when F is nonconvex valued (“nonconvex problem”). We also show the existence of extremal trajectories, that is, solutions of Eq. 1.1 when F(t, x) is replaced by ext F(t, x) (the set of extreme points of F(t, x)). Moreover, we show that every solution of the convex problem can be approximated in the \(C(T;\mathbb {R}^{N})\)-norm by certain extremal trajectories (“strong relaxation” theorem). An example of a feedback periodic control system illustrate our results.
Our work here is related to those of Frigon [5] and Qin and Xue [14]. In Frigon [5], φ ≡ 0 and A is time-independent with \(D(A)\ne \mathbb {R}^{N}\). The author proves existence theorems for both the convex and nonconvex problems using the notion of Lp-solution tube. Qin and Xue [14] assume that that A is time-independent and is a positive definite N × N-matrix. Also, they assume that \(\varphi \colon \mathbb {R}^{N}\longrightarrow \mathbb {R}\) is continuous convex. They deal with the convex and nonconvex problems and also address the question of existence of extremal trajectories. Finally, we mention the work of Bader and Papageorgiou [1], where A ≡ 0, but the inclusion takes place in the context of a general separable Hilbert space.
2 Mathematical Background—Hypotheses
Our approach is based on tools from multivalued analysis (see Hu-Papageorgiou [11]) and from the theory of nonlinear operators of monotone type (see Gasiński-Papageorgiou [7] and Zeidler [15]).
Let (Ω,Σ) be a measurable space and X a separable Banach space. We use the following notation:
A multifunction (set-valued function), F : Ω→2X ∖∅ is said to be “graph measurable,” if \(\text {Gr}\ F\in {\Sigma }\otimes \mathcal {B}(X)\), where
with \(\mathcal {B}(X)\) being the Borel σ-field of X. If \({\Sigma }=\widehat {{\Sigma }}\), the universal σ-field (this is the case if there is a σ-finite measure μ on Σ and Σ is μ-complete), then the Yankov-von Neumann-Aumann selection theorem (see Hu and Papageorgiou [11, p.158] or Gasiński and Papageorgiou [7, p. 906]) says that every graph measurable multifunction F : Ω→2X ∖∅ admits a measurable selection, that is, there exists a Σ-measurable function f : Ω→X such that f(ω) ∈ F(ω) for all ω ∈Ω. In fact, there is a whole sequence \(\{f_{n}\colon {\Omega }\longrightarrow X\}_{n\geqslant 1}\) of measurable selections such that
The result is true if the separable Banach space X is replaced by a Souslin space. Recall that a Souslin space is always separable but need not be metrizable. For example, if X∗ is the dual of a separable Banach space and it is equipped with the w∗-topology, then it is a nonmetrizable Souslin space.
A multifunction \(F\colon {\Omega }\longrightarrow P_{f}(X)\) is said to be “measurable”, if for all x ∈ X, the function
is Σ-measurable. This is equivalent to saying that for every open set \(U\subseteq X\), the set
A measurable multifunction F : Ω→Pf(X) is graph measurable. The converse is true if there is a σ-finite, complete measure defined on Σ.
Now, let (Ω,Σ, μ) be a σ-finite measure space and X a separable Banach space. Given \(1\leqslant p\leqslant +\infty \) and a graph measurable multifunction F : Ω→2X ∖∅, we define
A straightforward application of the Yankov-von Neumann-Aumann selection theorem, reveals that “\({S_{F}^{p}}\ne \emptyset \) if and only if \(\omega \longmapsto \inf \{\|u\|:\ u\in F(\omega )\}\) belongs in Lp(Ω).” This set is “decomposable,” that is, if \((A,f_{1},f_{2})\in {\Sigma }\times {S_{F}^{p}}\times {S_{F}^{p}}\), then
Here, for C ∈Σ, χC denotes the characteristic function of C, hence
Since χω∖C = 1 − χC, we see that the notion of decomposability formally looks very similar to that of convexity, only now the coefficients in the linear combination are functions. In fact, decomposable sets exhibit properties which are similar the those of convex sets (see Hu and Papageorgiou [11, Section 2.3]).
Suppose that Y and Z are Hausdorff topological spaces and let G: Y →2Z ∖∅ be a multifunction. We say that G is “upper semicontinuous” if for every open set \(U\subseteq Z\), the set \(G^{+}(U)=\{y\in Y:\ G(y)\subseteq U\}\) is open. We say that G is “lower semicontinuous” if for every open set \(U\subseteq Z\), the set G−(U) = {y ∈ Y : G(y) ∩ U≠∅} is open.
An upper semicontinuous multifunction with closed values has closed graph. The converse is true, if G is locally compact (that is, for every x ∈ X, there exists a neighborhood U of x such that \(\overline {G(U)}\in P_{k}(Z)\)). If Z is a metric space, then G is lower semicontinuous if and only if for all z ∈ Z, \(y\longmapsto d_{Z}(z,G(y))=\inf \limits _{v\in G(y)}d_{Z}(z,v)\) is an upper semicontinuous \(\mathbb {R}_{+}\)-valued function (here, dZ denotes the metric of Z).
For a metric space Z (with metric dZ) on Pf(Z), we can define a generalized metric, known as the “Hausdorff metric”, by setting
If Z is complete, then so is (Pf(Z), h). A multifunction G: Y →Pf(Z) is said to be “h-continuous”, if it is continuous from the Hausdorff topological space Y into the metric space (Pf(Z), h).
Suppose that V is a Banach space and \(C\subseteq V\) is nonempty. We set
Also, if \(\{C_{n}\}_{n\geqslant 1}\subseteq 2^{V}\setminus \emptyset \), we define
and
Next, let X be a reflexive Banach space and X∗ its topological dual. By 〈⋅,⋅〉, we denote the duality brackets for the pair (X∗, X). A multivalued map \(A\colon X\supseteq D\longrightarrow 2^{X^{*}}\) is said to be “monotone”, if
Here, D = {x ∈ X : A(x)≠∅}, the “domain” of A.
We say that a monotone map is “strictly monotone,” if
The monotone map \(A\colon X\supseteq D\longrightarrow 2^{X^{*}}\) is “maximal monotone,” if
This means that Gr A is maximal with respect to inclusion among the graphs of monotone maps. It is easy to see that, if \(A\colon X\supseteq D\longrightarrow 2^{X^{*}}\) is maximal monotone, then Gr A is sequentially closed in Xw × X∗ and in \(X\times X_{w}^{*}\) (here, by Xw and \(X_{w}^{*}\), we denote the spaces X and X∗, respectively, furnished with the weak topology). If A is maximal monotone, then for every x ∈ D, A(x) ∈ Pfc(X∗).
For a maximal monotone map \(A\colon X\longrightarrow 2^{X^{*}}\), we define
Since for every x ∈ D, A(x) ∈ Pfc(X∗) and X is reflexive, then A0(x)≠∅. Moreover, if X∗ is strictly convex, then A0 is single-valued. The map A0 is called the “minimal section” of A.
The “duality map” \(\mathcal {F}\colon X\longrightarrow 2^{X^{*}}\) is defined by
The Hahn-Banach theorem implies that \(\mathcal {F}\) has nonempty values. In fact, the duality map is defined for any Banach space. However, its properties strongly depend on the geometry of the Banach space X. In particular, if X and X∗ are both locally uniformly convex, then \(\mathcal {F}\) is single-valued and a homeomorphism.
By Γ0(X), we denote the cone of all functions \(\varphi \colon X\longrightarrow \overline {\mathbb {R}}=\mathbb {R}\cup \{+\infty \}\) which are lower semicontinuous, convex, and proper (that is, \(\text {dom}\ \varphi =\{x\in X:\ \varphi (x)<\infty \}\) (the effective domain of φ) is nonempty). By \(\partial \varphi \colon X\longrightarrow 2^{X^{*}}\), we denote the subdifferential of φ in the sense of convex analysis, that is
If φ is continuous at u, then ∂φ(u)≠∅. If φ is Gâteaux differentiable at u, then \(\partial \varphi (u)=\{\varphi _{G}^{\prime }(u)\}\) (\(\varphi _{G}^{\prime }(u)\) denotes the Gâteaux derivative of φ at u). The map \(\partial \varphi \colon X\longrightarrow 2^{x^{*}}\) is maximal monotone.
By \({L^{1}_{w}}(T;\mathbb {R}^{N})\), we denote the Lebesgue space \(L^{1}(T;\mathbb {R}^{N})\) equipped with the weak norm
or equivalently by
This norm is equivalent to the Pettis norm (see Egghe [4]).
The hypotheses on the map A and on the function φ are the following:
H(A): \(A\colon T\times \mathbb {R}^{N}\longrightarrow 2^{\mathbb {R}^{N}}\setminus \emptyset \) is a multifunction such that 0 ∈ A(t,0) for all t ∈ T and
- (i):
-
(t, x)→A(t, x) is graph measurable and for all t ∈ T, x↦A(t, x) is maximal monotone.
- (ii):
-
There exist two continuous functions \(\eta \colon T\longrightarrow \mathbb {R}^{N}\) and \(l\colon \mathbb {R}_{+}\longrightarrow \mathbb {R}_{+}\) such that
$$(h_{1}-h_{2},u_{1}-u_{2})_{\mathbb{R}^{N}}\geqslant -|\eta(t)-\eta(s)||u_{1}-u_{2}|l(\max\{|u_{1}|,|u_{2}|\}) $$for all \(0\leqslant s\leqslant t\leqslant b\) and all (u1, h1) ∈Gr A(t,⋅), (u2, h2) ∈Gr A(s,⋅).
- (iii):
-
For every r > 0, there exists ar ∈ L2(T) such that
$$|A(t,x)|\leqslant a_{r}(t)\quad\textrm{for a.a.}\ t\in T,\ \text{all}\ |x|\leqslant r $$and for all \(u\in L^{2}(T;\mathbb {R}^{N})\), t↦A0(t, u(t)) belongs in \(L^{2}(T;\mathbb {R}^{N})\).
Remark 2.1
Hypothesis H(A)(i) implies that
Hypothesis H(A)(ii) imposes restrictions on the t-dependence of A and permits the use of the theory of evolution equations involving time-dependent operators (see Pavel [13]).
H(φ): \(\varphi \in {\Gamma }_{0}(\mathbb {R}^{N})\) with 0 ∈ D(∂φ) and
Remark 2.2
Both conditions in the above hypothesis imply that u↦∂φ(u) is bounded (that is, maps bounded sets to bounded sets). The following function φ satisfies the first condition, namely that D(∂φ) = intdom φ
for all \(u\in \mathbb {R}^{N}\).
Next, we prove a result which we will need in the sequel and which is of independent interest. For this reason, it is formulated in a more general setting than the one in which it will be used in this paper. We mention that the result is known for Hilbert spaces (see Brézis [3, p. 25]).
So, as before, let X be a reflexive Banach space and X∗ its topological dual. By |⋅| and |⋅|∗, we denote the norm on X and X∗ respectively and by 〈⋅,⋅〉 the duality brackets for the pair (X∗, X). Let \(A\colon X\longrightarrow 2^{X^{*}}\) be a maximal monotone map with 0 ∈ A(0). On account of the Troyanski renorming theorem (see, e.g., Gasiński-Papageorgiou [7, p. 911]), without any loss of generality, we may assume that both X and X∗ are locally uniformly convex. We know that the duality map \(\mathcal {F}\colon X\longrightarrow X^{*}\) is single-valued and a homeomorphism. We introduce the lifting (realization) of A on the dual pair \((L^{p^{\prime }}(T;X^{*}),L^{p}(T;X))\), \(1\leqslant p<+\infty \), \(1<p^{\prime }\leqslant +\infty \), \(\frac {1}{p}+\frac {1}{p^{\prime }}= 1\) (recall that \(L^{p}(T;X)^{*}=L^{p^{\prime }}(T;X^{*})\); see Gasiński and Papageorgiou [7, p. 129]), \(\mathcal {A}\colon L^{p}(T;X)\longrightarrow L^{p^{\prime }}(T;X^{*})\) defined by
for all \(u\in D(\mathcal {A})=\{u\in L^{p}(T;X):\ S_{A(u(\cdot ))}^{p^{\prime }}\ne \emptyset \}\).
In what follows by ((⋅,⋅)), we denote the duality brackets for the pair of spaces \((L^{p^{\prime }}(T;X^{*}),L^{p}(T;X))\). So, we have
Lemma 2.3
If \(A\colon X\longrightarrow 2^{X^{*}}\)isa maximal monotone map with 0 ∈ A(0),then \(\mathcal {A}\colon L^{p}(T;X)\longrightarrow 2^{L^{p^{\prime }}(T;X^{*})}\)ismaximal monotone.
Proof
Let \(\vartheta \colon L^{p}(T;X)\longrightarrow L^{p^{\prime }}(T;X^{*})\) be the map defined by
Evidently, 𝜗 is continuous and strictly monotone, thus, maximal monotone too (see Gasiński and Papageorgiou [7, p. 310]).
- Claim :
-
\(R(\mathcal {A}+\vartheta )=L^{p^{\prime }}(T;X^{*})\) (that is, \(\mathcal {A}+\vartheta \) is surjective).
Let \(h\in L^{p^{\prime }}(T;X^{*})\) and consider the multifunction K : T→2X defined by
The map \(x\longmapsto A(x)+|x|^{p-2}\mathcal {F}\) is maximal monotone and coercive. Hence, it is surjective (see Gasiński and Papageorgiou [7, p. 336]). Therefore, K(t)≠∅ for all t ∈ T ∖ N, with N being Lebesgue-null. On this exceptional null set, we set K(t) = {0}. Note that
We know that the maximal monotonicity of A implies that \(\text {Gr}\ A\subseteq X\times X^{*}\) is closed. Moreover, the map ξ: T × X→X × X∗ defined by
is a Carathéodory mao, that is, for all x ∈ X, t↦ξ(t, x) is measurable, while for almost all t ∈ T, x↦ξ(t, x) is continuous. We know that ξ is jointly measurable (see Hu and Papageorgiou [11, p. 142]). Hence
with \(\mathcal {L}_{T}\) being the Lebesgue σ-field on T and \(\mathcal {B}(X)\) the Borel σ-field of X. Invoking the Yankov-von Neumann-Aumann selection theorem, we can find a measurable map u: T→X such that
so
We act with u(t) ∈ X and recall that by hypothesis 0 ∈ A(0), we obtain
thus, u ∈ Lp(T;X) (recall that \(p-1=\frac {p}{p^{\prime }}\), \(h\in L^{p^{\prime }}(T;X^{*})\)) and \(h\in \mathcal {A}(u)+\vartheta (u)\).
Since \(h\in L^{p^{\prime }}(T;X^{*})\) is arbitrary, we conclude that
This proves the Claim.
Evidently the map \(\mathcal {A}\) is monotone. We will show that in fact it is maximal monotone. To this end, suppose that \((v,g)\in L^{p}(T;X)\times L^{p^{\prime }}(T;X^{*})\) and assume that
On account of the Claim, we can find \((u_{1},h_{1})\in \text {Gr}\ \mathcal {A}\) such that
We return to Eq. 2.1 and choose (u, h) = (u1, h1). Then, using Eq. 2.2, we have
so u1 = v (since 𝜗 is strictly monotone), thus, (v, g) ∈Gr A and hence \(\mathcal {A}\) is maximal monotone. □
3 The Convex Problem
In this section, we prove an existence theorem for problem (1.1) when the multivalued perturbation F is convex valued.
The precise hypotheses on F are the following:H(F)1: \(F\colon T\times \mathbb {R}^{N}\longrightarrow P_{kc}(\mathbb {R}^{N})\) is a multifunction such that
- (i):
-
For all \(x\in \mathbb {R}^{N}\), t→F(t, x) is graph measurable.
- (ii):
-
For almost all t ∈ T, \(\text {Gr}\ F(t,\cdot )\in \mathbb {R}^{N}\times \mathbb {R}^{N}\) is closed.
- (iii):
-
There exist M > 0 and \(\widehat {a}_{M}\in L^{2}(T)\) such that
$$0\leqslant (h,x)_{\mathbb{R}^{N}}\quad\textrm{for a.a.}\ t\in T,\ \text{all}\ |x|=M,\ h\in F(t,x), $$$$|F(t,x)|\leqslant \widehat{a}_{M}(t)\quad\textrm{for a.a.}\ t\in T,\ |x|\leqslant M. $$
Remark 3.1
Hypotheses H(F)1(i) and (ii) do not imply joint measurability of F (see Hu and Papageorgiou [11, p. 226]). So, for \(u\colon T\longrightarrow \mathbb {R}^{N}\) measurable, \(|u(t)|\leqslant M\) almost everywhere on T, it is not a priori clear that \(S_{F(\cdot ,u(\cdot ))}^{2}\ne \emptyset \). To show the nonemptiness of this set, we argue as follows. Let \(\{s_{n}\}_{n\geqslant 1}\) be a sequence of step functions such that sn(t)→u(t) for almost all t ∈ T and \(|s_{n}(t)|\leqslant |u(t)|\) for almost all t ∈ T, all \(n\in \mathbb {N}\). Then, hypothesis H(F)1 implies that for every \(n\in \mathbb {N}\), t↦F(t, sn(t)) is measurable and so by the Yankov-von Neumann-Aumann selection theorem, we can find \(h_{n}\colon T\longrightarrow \mathbb {R}^{N}\) measurable such that
so the sequence \(\{h_{n}\}_{n\geqslant 1}\subseteq L^{2}(T;\mathbb {R}^{N})\) is bounded.
Therefore, by passing to a subsequence if necessary, we may assume that
with \(h\in L^{2}(T;\mathbb {R}^{N})\). Invoking Proposition 3.9 of Hu and Papageorgiou [11, p. 694], we have
(see hypothesis H(F)1(ii)), so \(h\in S_{F(\cdot ,u(\cdot ))}^{2}\). Hypothesis H(F)1(ii) is a multivalued variant of a condition first used by Hartman [10].
Together with H(F)1, we will need the following extra condition on ∂φ:
- H0::
-
For all x ∈ D(∂φ) and g ∈ ∂φ(x), we have \((g,x)_{\mathbb {R}^{N}}\geqslant 0\).
Alternatively, instead of H(F)1, H0, we can use the following conditions on F:\(H(F)_{1}^{\prime }\): \(F\colon T\times \mathbb {R}^{N}\longrightarrow P_{kc}(\mathbb {R}^{N})\) is a multifunction such that hypotheses \(H(F)_{1}^{\prime }(i)\) and (ii) are the same as the corresponding hypotheses H(F)1(i) and (ii) and
- (iii):
-
\(|F(t,x)|\leqslant k(t)(1+|x|)|\) for almost all t ∈ T, all \(x\in \mathbb {R}^{N}\), with k ∈ L2(T).
We know that
(see Hu and Papageorgiou [11, p.346]). Let \(x_{0}\in \overline {\text {dom}\ \varphi }\), \(h\in L^{2}(T;\mathbb {R}^{N})\) and ε > 0. We consider the following auxiliary Cauchy problem:
We have the following existence and uniqueness theorem for this problem.
Proposition 3.2
If hypothesesH(A) andH(φ) hold,then problem (3.1) admits a unique solution\(u_{0}\in W^{1,2}((0,b);\mathbb {R}^{N})\subseteq C(T;\mathbb {R}^{N})\).
Proof
Let
For all t ∈ T, V (t,⋅) is maximal monotone. Moreover, because of the monotonicity of x↦∂φ(x) and of x↦εx, V (t, x) satisfies hypothesis H(A)(ii). Therefore, we can apply Theorem 1.2 of Pavel [13] and have a solution \(u_{0}\in C(T;\mathbb {R}^{N})\) of problem (3.1). We have \(u_{0}(t)\in D(\partial \varphi )\) for all t ∈ T and recall that on account of hypothesis H(φ), ∂φ maps bounded sets to bounded sets. Therefore,
for some M > 0. This fact together with hypothesis H(A)(iii) implies that \(u_{0}^{\prime }\in L^{2}(T;\mathbb {R}^{N})\), so \(u_{0}\in W^{1,2}((0,b);\mathbb {R}^{N})\).
Next, we show that this solution is unique. So, suppose that \(u_{0},v_{0}\in W^{1,2}((0,b);\mathbb {R}^{N})\) are two such solutions. We have
with u0(0) = v0(0) = x0.
Subtracting (3.3) from Eq. 3.2, we obtain
We take inner product with u0(t) − v0(t). The monotonicity of A(t,⋅) and ∂φ implies that
so
(since u0(0) = v0(0) = x0), hence u0 = v0. This proves the uniqueness of the solution of problem (3.1). □
We consider the Poincaré map \(P\colon \overline {\text {dom}\ \varphi }\longrightarrow \overline {\text {dom}\ \varphi }\) defined by
with \(u_{0}\in W^{1,2}((0,b);\mathbb {R}^{N})\) being the unique solution of Eq. 3.1 (see Proposition 3.2).
Proposition 3.3
If hypothesesH(A) andH(φ) hold,then the PoincarémapP is a contraction.
Proof
Let \(x_{0},\widehat {x}\in \overline {\text {dom}\ \varphi }\) be two distinct initial conditions for problem (3.1) and let \(u_{0},\widehat {u}\in W^{1,p}((0,b);\mathbb {R}^{N})\) be the corresponding unique solutions of the Cauchy problem. We have
As in the proof of Proposition 3.2, we subtract (3.6) from (3.5) and then take inner product with \(u_{0}(t)-\widehat {u}(t)\) to obtain
so
thus
Let t = b. Then,
so P is a contraction. □
Now, for \(h\in L^{2}(T;\mathbb {R}^{N})\), we consider the following auxiliary periodic system:
Proposition 3.4
If hypothesesH(A) andH(φ) hold,then problem (3.7) admits a unique solution\(u_{0}\in W^{1,2}((0,b);\mathbb {R}^{N})\subseteq C(T;\mathbb {R}^{N})\)andwe have
for some c0 > 0.
Proof
From Proposition 3.3, we know that the Poincaré map \(P\colon \overline {\text {dom}\ \varphi }\longrightarrow \overline {\text {dom}\ \varphi }\) is a contraction. So, by the Banach fixed point theorem, there is a unique \(u_{0}\in \overline {\text {dom}\ \varphi }\) such that
The corresponding solution \(u_{0}\in W^{1,2}((0,b);\mathbb {R}^{N})\) of Eq. 3.1 is the unique solution of the periodic system (3.7).
Then on account of hypothesis H(A)(iii), we can find \(g_{0}\in S_{\partial \varphi (u_{0}(\cdot ))}^{2}\) such that
We act with u0(t) and recall that 0 ∈ A(t,0) for all t ∈ T, we obtain
for almost all t ∈ T, all v0 ∈ ∂φ(0), so
thus,
It follows that
Note that
thus,
In Eq. 3.9, we choose t = b. Using the periodic boundary condition, we have
so
We return to Eq. 3.9 and use Eq. 3.10. Then,
so
for some c0 > 0. □
Let M > 0 be as postulated by hypothesis H(F)1(iii) and let \(p_{M}\colon \mathbb {R}^{N}\longrightarrow \mathbb {R}^{N}\) be the M-radial retraction defined by
We know that pM is nonexpansive (that is, \(|p_{M}(x)-p_{M}(u)|\leqslant |x-u|\) for all \(x,u\in \mathbb {R}^{N}\)). We set
Evidently, \(\widehat {F}(t,x)\) satisfies hypotheses H(F)1(i) and (ii) and in addition, we have
In the sequel by \(\widehat {S}_{\varepsilon }\), we denote the solution set of the following periodic system
By Sε, we denote the solution set of
In the next proposition, we derive uniform a priori bounds for the elements of these two solution sets.
Proposition 3.5
- (a) :
-
If hypothesesH(A),H(φ),H(F)1,andH0hold,then \(|u(t)|\leqslant M\)forallt ∈ T,all \(u\in \widehat {S}_{\varepsilon }\)(here,M > 0 isas postulated in hypothesisH(F)1(iii)).
- (b) :
-
If hypothesesH(A),H(φ),and \(H(F)_{1}^{\prime }\)hold,then there existsM > 0 suchthat\(|u(t)|\leqslant M\)forallt ∈ T,allu ∈ Sε.
Proof
(a) Let \(u\in \widehat {S}_{\varepsilon }\subseteq W^{1,2}((0,b);\mathbb {R}^{N})\). We have
with \(h\in S_{\widehat {F}(\cdot ,u(\cdot ))}^{2}\). Suppose that the result is not true. Then, two situations can occur:
- (I):
-
|u(t)| > M for all t ∈ T.
- (II):
-
There exist \(0\leqslant \eta \leqslant \tau \leqslant b\) such that
$$|u(\eta)|=M\ \text{and} \ |u(t)|>M \quad\forall t\in [\eta,\tau]. $$
Suppose that (I) holds. From Eq. 3.11 for some \(a\in S^{2}_{A(\cdot ,u(\cdot ))}\) and some \(g\in S^{2}_{\partial \varphi (u(\cdot ))}\), we have
We take inner product with u(t). Then,
for a.a. t ∈ T. Note that
Since \(a\in S^{2}_{A(\cdot ,u(\cdot ))}\) and 0 ∈ A(t,0) for all t ∈ T, we have
Hypothesis H0 implies that
Therefore, finally, we have
thus,
(see (I)). Note that
(see hypothesis H(F)1(iii)). Using this in Eq. 3.12, we obtain
a contradiction.
Next, suppose that (II) holds. Then, repeating the above argument on the interval [η, τ], we obtain
a contradiction.
So, we conclude that
(b) Let \(u\in S_{\varepsilon }\subseteq W^{1,2}((o,b);\mathbb {R}^{N})\). We have
with \(h\in S_{F(\cdot ,u(\cdot ))}^{2}\). Using Eq. 3.8 from Proposition 3.4, we have
(see hypothesis \(H(F)_{1}^{\prime }(iii)\)), so
for some M > 0 (using Gronwall’s inequality). □
On account of Proposition 3.5, we can always replace F(t, x) by \(\widehat {F}(t,x)=F(t,p_{M}(x))\). Therefore, without any loss of generality, we may assume that
with 𝜗 ∈ L2(T).
Proposition 3.4 implies that we can define the solution map \(\gamma _{\varepsilon }\colon L^{2}(T;\mathbb {R}^{N})\longrightarrow C(T;\mathbb {R}^{N})\) which to every \(h\in L^{2}(T;\mathbb {R}^{N})\) assigns the unique solution \(\gamma _{\varepsilon }(h)\in W^{1,2}((0,b);\mathbb {R}^{N})\subseteq C(T;\mathbb {R}^{N})\).
Proposition 3.6
If hypothesesH(A) andH(φ) hold,then the solution map \(\gamma _{\varepsilon }\colon L^{2}(T;\mathbb {R}^{N})\longrightarrow C(T;\mathbb {R}^{N})\)iscompletely continuous (that is, if \(h_{n} \overset {w}{\longrightarrow } h\)in \(L^{2}(T;\mathbb {R}^{N})\),thenγε(hn)→γε(h) in \(C(T;\mathbb {R}^{N})\)).
Proof
Let \(h_{n} \overset {w}{\longrightarrow } h\) in \(L^{2}(T;\mathbb {R}^{N})\) and set \(u_{n}=\gamma _{\varepsilon }(u_{n})\in W^{1,2}((0,b);\mathbb {R}^{N})\subseteq C(T;\mathbb {R}^{N})\) for all \(n\in \mathbb {N}\) and \(u=\gamma _{\varepsilon }(h)\in W^{1,2}((0,b);\mathbb {R}^{N})\subseteq C(T;\mathbb {R}^{N})\). We have
and
As before (see the proof of Proposition 3.2), subtracting (3.15) from Eq. 3.14 and taking inner product with un(t) − u(t), we obtain
From Eq. 3.8, we know that
so
with c1 > 0. Both conditions in hypothesis H(φ) imply that ∂φ maps bounded sets to bounded sets (see Gasiński-Papageorgiou [9, Problem 3.59]). Therefore,
for some c2 > 0 (see Eq. 3.17). Also, from hypothesis H(A)(iii), we have
with k0 ∈ L2(T). Returning to Eq. 3.14 and using Eqs. 3.17, 3.18, and 3.19, we infer that the sequence \(\{u_{n}^{\prime }\}_{m\geqslant 1}\subseteq L^{2}(T;\mathbb {R}^{N})\) is bounded, so the sequence \(\{u_{n}\}_{n\geqslant 1}\subseteq W^{1,2}((0,b);\mathbb {R}^{N})\) is bounded (see Eq. 3.17).
Thus, by passing to a suitable subsequence if necessary, we may assume that
In Eq. 3.16, we pass to the limit as \(n\to +\infty \) and use Eq. 3.20. Then,
Let t = b and recall that \(\widehat {u}(0)=\widehat {u}(b)\), u(0) = u(b). We obtain
so \(\widehat {u}=u\). Therefore, for the original sequence, we have un→u in \(C(T;\mathbb {R}^{N})\); thus, γε is completely continuous. □
On account of Eq. 3.13, we introduce the following set:
Evidently, \(W\subseteq L^{2}(T;\mathbb {R}^{N})\) is sequentially weakly compact (Eberlein-Smulian theorem; see Gasiński and Papageorgiou [7, p. 909]). Using Proposition 3.6, we have
so
(see Gasiński and Papageorgiou [8, p. 852, Theorem 5.86]).
Now, we can have our existence theorem for the “convex problem.”
Theorem 3.7
If hypothesesH(A),H(φ) andH(F)1,H0or \(H(F)_{1}^{\prime }\)hold,then problem (1.1) admits a solution \(u_{0}\in W^{1,2}((0,b);\mathbb {R}^{N})\).
Proof
Let ε > 0 and consider the following periodic system
Recall that F(t, x) satisfies (3.13). We consider the set \(W\subseteq L^{2}(T;\mathbb {R}^{N})\) furnished with the relative weak topology and introduce the multifunction Hε: W→Pkc(W) defined by
Let \(\{(g_{n},h_{n})\}_{n\geqslant 1}\subseteq \text {Gr}\ H_{\varepsilon }\) and assume that
Then Proposition 3.6 and Eq. 3.22 imply that
Using Proposition 3.9 of Hu and Papageorgiou [11, p. 694], we have
(see Eq. 3.23 and hypothesis H(F)1(ii)), so (g, h) ∈Gr Hε and thus Hε has closed graph in W × W.
The set W furnished with the relative weak topology is compact. Also, it is convex. Then, invoking the Kakutani-Ky Fan fixed point theorem (see Papageorgiou and Kyritsi [12, p. 114, Theorem 2.6.7] or Gasiński and Papageorgiou [7, p. 887]), we can find \(g_{0}^{\varepsilon }\in W\) such that \(g_{0}^{\varepsilon }\in H_{\varepsilon }(g_{0}^{\varepsilon })\), so \(g_{0}^{\varepsilon }\in S_{F(\cdot ,\gamma _{\varepsilon }(g_{0}^{\varepsilon })(\cdot ))}^{2}\).
Now, let εn→0+ and let
As before (see the proof of Proposition 3.6), using Eq. 3.8, we have that the sequence \(\{{u_{0}^{n}}\}_{n\geqslant 1}\subseteq C(T;\mathbb {R}^{N})\) is relatively compact.
Therefore, passing to a subsequence if necessary, we may assume that
From Eqs. 3.21 and 3.24 as before (see the proof of Proposition 3.6), we have that the sequence \(\{({u_{0}^{n}})'\}_{n\geqslant 1}\subseteq L^{2}(T;\mathbb {R}^{N})\) is bounded. Passing to a next subsequence if necessary, we may assume
For every \(n\in \mathbb {N}\) and every t, s ∈ T, s < t, we have
so
(see Eqs. 3.24 and 3.25), thus \(h=u_{0}^{\prime }\).
Let \({\Phi }\colon L^{2}(T;\mathbb {R}^{N})\longrightarrow \overline {\mathbb {R}}=\mathbb {R}\cup \{+\infty \}\) be the integral functional defined by
We know that
(see Hu and Papageorgiou [11, p. 349]). From Eq. 3.21, we have
where \(\mathcal {A}(u)\in S_{A(\cdot ,u(\cdot ))}^{2}\) for all \(u\in L^{2}(T;\mathbb {R}^{N})\). From Lemma 2.3, we have that \(\mathcal {A}\) is maximal monotone. In addition, hypothesis H(A)(iii) implies that \(D(\mathcal {A})=L^{2}(T;\mathbb {R}^{N})\). Therefore, Theorem 3.2.41 of Gasiński and Papageorgiou [7] implies that \(u\longmapsto (\mathcal {A}+\partial {\Phi })(u)\) is maximal monotone. Hence, its graph is sequentially closed in \(L^{2}(T;\mathbb {R}^{N})\times L^{2}(T;\mathbb {R}^{N})_{w}\). Note that \(\{{g_{0}^{n}}\}_{n\geqslant 1}\subseteq W\). So, we can say that
and \(g_{0}\in S_{F(\cdot ,u_{0}(\cdot ))}^{2}\) (see Eq. 3.24 and hypothesis H(F)1(ii)).
Therefore, if in Eq. 3.26 we pass to the limit as \(n\to +\infty \), then
(see Eq. 3.25), so
□
An interesting byproduct of the above proof is the following result concerning the solution set \(S_{c}\subseteq C(T;\mathbb {R}^{N})\) of the “convex problem.”
Proposition 3.8
If hypothesesH(A),H(φ) andH(F)1,H0,or \(H(F)_{1}^{\prime }\)hold,then \(S_{c}\in P_{k}(C(T;\mathbb {R}^{N}))\).
4 The Nonconvex Problem
In this section, we prove an existence theorem for the “nonconvex problem” (that is, the multivalued perturbation has nonconvex values).
In this case, the hypotheses on the multivalued perturbation F are the following:H(F)2: \(F\colon T\times \mathbb {R}^{N}\longrightarrow P_{f}(\mathbb {R}^{N})\setminus \emptyset \) is a multifunction such that
- (i):
-
(t, x)→F(t, x) is graph measurable.
- (ii):
-
For almost all t ∈ T, x↦F(t, x) is lower semicontinuous.
- (iii):
-
There exist M > 0 and \(\widehat {a}_{M}\in L^{2}(T)\) such that
$$0\leqslant (h,x)_{\mathbb{R}^{N}}\quad\textrm{for a.a.}\ t\in T,\ \text{all}\ |x|=M,\ h\in F(t,x), $$$$|F(t,x)|\leqslant \widehat{a}_{M}(t)\quad\textrm{for a.a.}\ t\in T,\ |x|\leqslant M. $$
As before (see Section 3), these hypotheses will be combined with H0. Alternatively, instead of the pair H(F)2, H0, we can use the following hypotheses on F:\(H(F)_{2}^{\prime }\): \(F\colon T\times \mathbb {R}^{N}\longrightarrow P_{f}(\mathbb {R}^{N})\) is a multifunction such that hypotheses \(H(F)_{2}^{\prime }(i)\) and (ii) are the same as the corresponding hypotheses H(F)2(i) and (ii) and
- (iii):
-
\(|F(t,x)|\leqslant k(t)(1+|x|)|\) for almost all t ∈ T, all \(x\in \mathbb {R}^{N}\), with k ∈ L2(T).
Theorem 4.1
If hypothesesH(A),H(φ) andH(F)2,H0,or \(H(F)_{2}^{\prime }\)hold,then problem (1.1) admits a solution \(u_{0}\in W^{1,2}((0,b);\mathbb {R}^{N})\subseteq C(T;\mathbb {R}^{N})\).
Proof
The a priori bounds in Proposition 3.5 remain valid and so without any loss of generality, we may assume that
with 𝜗 ∈ L2(T) (just replace F(t, x) with \(\widehat {F}(t,x)=F(t,p_{M}(x))\)).
Again, we introduce the sets
We introduce the multifunction \(V\colon C(T;\mathbb {R}^{N})\longrightarrow P_{wk}(L^{2}(T;\mathbb {R}^{N}))\) defined by
We show that V is a lower semicontinuous multifunction. According to Proposition 2.6 of Hu and Papageorgiou [11, p. 37], to show the lower semicontinuity of V, it suffices to show that, if un→u in C(T;H), then
So, suppose that
For \(n\in \mathbb {N}\), we consider the multifunction \(G_{n}\colon T\longrightarrow P_{k}(\mathbb {R}^{N})\) defined by
Hypothesis H(F)2(i) implies that the map t↦d(h(t), F(t, un(t))) is measurable. Therefore,
is a Carathéodory function. We know that Carathéodory functions are jointly measurable (see Hu and Papageorgiou [11, p. 142]). Therefore,
Invoking the Yankov-von Neumann-Aumann selection theorem, we can find a measurable function \(h_{n}\colon T\longrightarrow \mathbb {R}^{N}\) such that
We have
so
(see Hu and Papageorgiou [11, p. 672, Proposition 1.47], Eq. 4.2, hypothesis H(F)2(ii) and recall that h ∈ V (u)), thus,
Also, we have
(see Eq. 4.1), so
Then, from Eqs. 4.3 and 4.4 and Vitali’s theorem (see, e.g., Gasiński and Papageorgiou [7, p. 901]), we have
with hn ∈ V (un) for all \(n\in \mathbb {N}\), thus, V is lower semicontinuous.
Clearly, V has decomposable values. Applying the Bressan-Colombo-Fryszkowski selection theorem (see Bressan and Colombo [2], Fryszkowski [6], and Hu and Papageorgiou [11, p. 245, Theorem 8.7]), we can find a continuous map \(v\colon C(T;\mathbb {R}^{N})\longrightarrow L^{2}(T;\mathbb {R}^{N})\) such that
We set \(\tau _{\varepsilon }=\gamma _{\varepsilon }\circ v\colon K_{c}^{\varepsilon }\longrightarrow K_{c}^{\varepsilon }\). Evidently, τε is continuous (see Proposition 3.6). Since \(K_{c}^{\varepsilon }\in P_{kc}(C(T;\mathbb {R}^{N}))\), we can apply the Schauder fixed point theorem and produce \(u_{0}^{\varepsilon }\in K_{c}^{\varepsilon }\) such that
so \(u_{0}^{\varepsilon }\in W^{1,2}((0,b);\mathbb {R}^{N}))\) is a solution of problem (3.21).
Now, consider a sequence εn → 0+ and let \(u_{n}=u_{0}^{\varepsilon _{n}}\) for \(n\in \mathbb {N}\). From Eqs. 3.8 and 4.1, we see that there exists cε > 0 such that
This implies that Eqs. 3.18 and 3.19 are valid and hence the sequence \(\{u_{n}^{\prime }\}_{n\geqslant 1}\subseteq L^{2}(T;\mathbb {R}^{N})\) is bounded.
For all t, s ∈ T, \(s\leqslant t\), by the Cauchy-Schwarz inequality, we have
for some c4 > 0, so, the sequence \(\{u_{n}\}_{n\geqslant 1}\) is equicontinuous.
The Arzela-Ascoli theorem implies that the sequence \(\{u_{0}^{\varepsilon _{n}}\}_{n\geqslant 1}\subseteq C(T;\mathbb {R}^{N})\) is relatively compact.
So, passing to a subsequence if necessary, we may assume that
Exploiting the continuity of V and reasoning as in the last part of the proof of Theorem 3.7, in the limit as \(n\to +\infty \), we obtain
so \(u_{0}\in W^{1,2}((0,b);\mathbb {R}^{N})\subseteq C(T;\mathbb {R}^{N})\) is a solution of Eq. 1.1. □
5 Extremal Trajectories
In this section, we deal with the following version of problem (1.1):
Here, by ext F(t, x), we denote the set of extreme points of F(t, x). We know that even if F(t,⋅) has strong continuity properties, the multifunction x↦ext F(t, x) need not have any (see Hu and Papageorgiou [11, Section 2.4]). So, the existence of solutions for problem (5.1) cannot be deduced from Theorems 3.7 and 4.1 and a different approach is needed.
We need to strengthen the conditions on the multivalued perturbation F(t, x). The new hypotheses are the following:H(F)3: \(F\colon T\times \mathbb {R}^{N}\longrightarrow P_{kc}(\mathbb {R}^{N})\) is a multifunction such that
- (i):
-
For all \(x\in \mathbb {R}^{N}\), t→F(t, x) is graph measurable.
- (ii):
-
For almost all t ∈ T, x↦F(t, x) is h-continuous.
- (iii):
-
There exist M > 0 and \(\widehat {a}_{M}\in L^{2}(T)\) such that
$$0\leqslant (h,x)_{\mathbb{R}^{N}}\quad\textrm{for a.a.}\ t\in T,\ \text{all}\ |x|=M,\ h\in F(t,x), $$$$|F(t,x)|\leqslant \widehat{a}_{M}(t)\quad\textrm{for a.a.}\ t\in T,\ |x|\leqslant M. $$
As before, H(F)3 will be combined with H0. Alternatively, we can replace the pair H(F)3, H0 with the following hypotheses:\(H(F)_{3}^{\prime }\): \(F\colon T\times \mathbb {R}^{N}\longrightarrow P_{kc}(\mathbb {R}^{N})\setminus \emptyset \) is a multifunction such that hypotheses \(H(F)_{3}^{\prime }(i)\) and (ii) are the same as the corresponding hypotheses H(F)3(i) and (ii) and
- (iii):
-
\(|F(t,x)|\leqslant k(t)(1+|x|)|\) for almost all t ∈ T, all \(x\in \mathbb {R}^{N}\), with k ∈ L2(T).
Theorem 5.1
If hypothesesH(A),H(φ) andH(F)3,H0,or \(H(F)_{3}^{\prime }\)hold,then problem (5.1) admits a solution \(u_{0}\in W^{1,2}((0,b);\mathbb {R}^{N})\subseteq C(T;\mathbb {R}^{N})\).
Proof
Given ε ∈ (0,1], first we consider the following regularization of Eq. 5.1:
The a priori bound from Proposition 3.5 remains valid. So, without any loss of generality we assume that
with 𝜗 ∈ L2(T). As before
while from the proof of Proposition 3.4 (see Eq. 3.10), we see that we can find \(\widehat {K}_{c}\in P_{kc}(C(T;\mathbb {R}^{N}))\) such that \(\overline {\text {conv}}\,\gamma _{\varepsilon }(W)\subseteq \widehat {K}_{c}\) for all ε ∈ (0,1].
Applying Theorem 8.31 of Hu and Papageorgiou [11, p. 260], we can find a continuous map \(\tau \colon \widehat {K}_{c}\longrightarrow {L^{1}_{w}}(T;\mathbb {R}^{N})\) such that
(see Hu and Papageorgiou [11, p. 191, Theorem 4.5]). Then, we consider the map \(\sigma _{\varepsilon }=\gamma _{\varepsilon }\circ \tau \colon \widehat {K}_{c}\longrightarrow \widehat {K}_{c}\). We claim that σε is continuous. So, let \(\{u_{n}\}_{n\geqslant 1}\subseteq \widehat {K}_{c}\) and assume that
so
Because of Eq. 5.3, we can apply Lemma 2.8 of Hu and Papageorgiou [11, p. 24] and have
thus,
thus, σε is continuous.
Since \(\sigma _{\varepsilon }\colon \widehat {K}_{c}\longrightarrow \widehat {K}_{c}\) and \(\widehat {K}_{c}\in P_{kc}(C(T;\mathbb {R}^{N}))\), the Schauder fixed point theorem gives \(u_{0}^{\varepsilon }\in \widehat {K}_{c}\) such that \(u_{0}^{\varepsilon }=\sigma _{\varepsilon }(u_{0}^{\varepsilon })\).
Let εn ∈ (0,1] for \(n\in \mathbb {N}\) and assume that εn → 0+. We set \({u_{0}^{n}}=u^{\varepsilon _{n}}_{0}\) for \(n\in \mathbb {N}\). From the proof of Theorem 4.1, we have that the sequence \(\{{u_{0}^{n}}\}_{n\geqslant 1}\subseteq C(T;\mathbb {R}^{N})\) is relatively compact. So, passing to a subsequence if necessary, we may assume that
For every \(n\in \mathbb {N}\), we have
Since \(\tau ({u_{0}^{n}}) \overset {w}{\longrightarrow } \tau (u_{0})\) in \(L^{2}(T;\mathbb {R}^{N})\) (see Lemma 2.8 of Hu and Papageorgiou [11, p. 24]) as in the proof of Theorem 3.7, in the limit as \(n\to +\infty \), we obtain
(see Eq. 5.4). Therefore, \(u_{0}\in W^{1,2}((0,b);\mathbb {R}^{N})\subseteq C(T;\mathbb {R}^{N})\) is a solution of Eq. 5.1. □
6 Strong Relaxation
In this section, we show that every solution of the convex problem can be obtained as the limit in the \(C(T;\mathbb {R}^{N})\)-norm of certain extremal trajectories. Such a result is known as “strong relaxation.” The result is important in many applications. In the context of control systems, it says that we can approximate any state of the system by states which are generated using “bang-bang controls.” This way, we can economize in the use of controls. In the context of game theory, the selection of ext F(⋅, u(⋅)) are known as “pure strategies” and the strong relaxation theorem implies that any state can be approximated by ones generated using only pure strategies.
To prove such a result, we need to strengthen further the conditions on F. The new hypotheses are the following:H(F)4: \(F\colon T\times \mathbb {R}^{N}\longrightarrow P_{kc}(\mathbb {R}^{N})\) is a multifunction such that
- (i):
-
For all \(x\in \mathbb {R}^{N}\), t→F(t, x) is graph measurable.
- (ii):
-
For every r > 0, there exists ηr ∈ L1(T) such that
$$h(F(t,x),F(t,y))\leqslant \eta_{r}(t)|x-y| $$for almost all t ∈ T, all \(x,y\in \mathbb {R}^{N}\) with \(|x|,|y|\leqslant r\).
- (iii):
-
Tthere exist M > 0 and \(\widehat {a}_{M}\in L^{2}(T)\) such that
$$0\leqslant (h,x)_{\mathbb{R}^{N}}\quad\textrm{for a.a.}\ t\in T,\ \text{all}\ |x|=M,\ h\in F(t,x), $$$$|F(t,x)|\leqslant \widehat{a}_{M}(t)\quad\textrm{for a.a.}\ t\in T,\ |x|\leqslant M. $$
These hypotheses go together with H0. Alternatively, the pair H(F)4, H0 can be replaced by the following hypotheses:\(H(F)_{4}^{\prime }\): \(F\colon T\times \mathbb {R}^{N}\longrightarrow P_{kc}(\mathbb {R}^{N})\setminus \emptyset \) is a multifunction such that hypotheses \(H(F)_{4}^{\prime }(i)\) and (ii) are the same as the corresponding hypotheses H(F)4(i) and (ii) and
- (iii):
-
\(|F(t,x)|\leqslant k(t)(1+|x|)|\) for almost all t ∈ T, all \(x\in \mathbb {R}^{N}\), with k ∈ L2(T).
In what follows by Sc, we denote the solution set of the convex problem (that is, in Eq. 1.1, F(⋅,⋅) has values in \(P_{kc}(\mathbb {R}^{N})\)). From Proposition 3.8, we know that \(S_{c}\in P_{k}(C(T;\mathbb {R}^{N}))\).
Let \(x_{0}\in \mathbb {R}^{N}\) and consider the following Cauchy problem:
Let \(S_{e}(x_{0})\subseteq W^{1,p}((0,b);\mathbb {R}^{N})\) be the solution set of Eq. 6.1. A simplified version of the proof of Theorem 5.1 shows that Se(x0)≠∅. Then, our strong relaxation result reads as follows.
Theorem 6.1
If hypothesesH(A),H(φ) andH(F)4,H0,or \(H(F)_{4}^{\prime }\)holdandu ∈ Sc,then there exists a sequence \(\{u_{n}\}_{n\geqslant 1}\subseteq S_{e}(u(0))\)suchthatun→uin \(C(T;\mathbb {R}^{N})\).
Proof
From the previous work, we know that without any loss of generality, we may assume that
with 𝜗 ∈ L2(T).
We know that \(S_{c}\in P_{k}(C(T;\mathbb {R}^{N}))\). Similarly, \(\overline {S_{e}(u(0))}^{C(T;H)}\in P_{k}(C(T;\mathbb {R}^{N}))\). We set
Since u ∈ Sc, we can find \(h\in S_{F(\cdot ,u(\cdot ))}^{2}\) such that
Given v ∈ K∗ and ε > 0, we introduce the multifunction \(L_{v,\varepsilon }\colon T\longrightarrow 2^{\mathbb {R}^{N}}\setminus \emptyset \) defined by
with \(h\in L^{2}(T;\mathbb {R}^{N})\) as in Eq. 6.3 and M0 = |K∗|. Hypotheses H(F)4(i) and (ii) imply that (t, x)↦F(t, x) is graph measurable. So, t↦F(t, v(t)) is measurable. Therefore, (t, y)↦ξ(t, y) = |h(t) − y|− d(h(t), F(t, v(v))) is a Carathéodory function. We have
so \(\text {Gr}\ L_{v,\varepsilon }\in \mathcal {L}_{T}\otimes \mathcal {B}(\mathbb {R}^{N})\). Invoking the Yankov-von Neumann-Aumann selection theorem, we can find a measurable function \(l_{v,\varepsilon }\colon T\longrightarrow \mathbb {R}^{N}\) such that
hence,
(see Eq. 6.2). Then, we introduce the multifunction \(H_{\varepsilon }\colon K^{*}\longrightarrow 2^{L^{2}(T;\mathbb {R}^{N})}\) defined by
From Eq. 6.4, we see that
Also Lemma 8.3 of Hu and Papageorgiou [11, p. 239] implies that v↦Hε(v) is lower semicontinuous and so \(v\longmapsto \overline {H_{\varepsilon }(v)}\) is lower semicontinuous (see Hu and Papageorgiou [11, p. 50]). Clearly, \(\overline {H_{\varepsilon }(\cdot )}\) has decomposable values. Hence, the selection theorem of Bressan-Colombo-Fryszkowski (see Bressan and Colombo [2] and Fryszkowski [6]) gives a continuous map \(\tau _{\varepsilon }\colon K^{*}\longrightarrow L^{2}(T;\mathbb {R}^{N})\) satisfying
Moreover, invoking Theorem 8.31 of Hu and Papageorgiou [11, p. 260], we can find a continuous map \(r_{\varepsilon }\colon K^{*}\longrightarrow {L^{1}_{w}}(T;\mathbb {R}^{N})\) such that
Now, let \(\varepsilon _{n}=\frac {1}{n}\), \(\tau _{n}=\tau _{\varepsilon _{n}}\), and \(r_{n}=r_{\varepsilon _{n}}\) for all \(n\in \mathbb {N}\) and \(x_{0}=u(0)=u(b)\). We consider the following Cauchy problem:
Let \(u_{n}\in W^{1,2}((0,b);\mathbb {R}^{N})\) be a solution of Eq. 6.7. Evidently, \(\{u_{n}\}_{n\geqslant 1}\subseteq S_{e}(x_{0})\) (see Eq. 6.6). Then, we have that the sequence \(\{u_{n}\}_{n\geqslant 1}\subseteq W^{1,2}((0,b);\mathbb {R}^{N})\) is bounded, so the sequence \(\{u_{n}\}_{n\geqslant 1}\subseteq C(T;\mathbb {R}^{N})\) is relatively compact. Passing to a subsequence if necessary, we may assume that
Exploiting the monotonicity of A(t,⋅) and of ∂φ and recalling that un(0) = x0 = u(0) for all \(n\in \mathbb {N}\) (see Eq. 6.7), we obtain
(see Eq. 6.5, hypotheses H(F)4(ii) and \(H(F)_{4}^{\prime }(ii)\)). From Eq. 6.6, we have
Then, Lemma 2.8 of Hu and Papageorgiou [11, p. 24] implies that
We return to Eq. 6.9, pass to the limit as \(n\to +\infty \) and use Eqs. 6.8 and 6.20. Then,
so
(by Gronwall’s inequality). It follows that \(\widehat {u}=u\). Therefore, \(u=\lim \limits _{n\rightarrow +\infty } u_{n}\) in \(C(T;\mathbb {R}^{N})\) and un ∈ Se(u(0)) for all \(n\in \mathbb {N}\). □
Let \(S_{e}\subseteq W^{1,2}((0,b);\mathbb {R}^{N})\subseteq C(T;\mathbb {R}^{N})\) be the solution set of problem (5.1). From Theorem 3.7, we know that Se≠∅. If we strengthen the conditions on A(t,⋅), we can show that the set Se is dense in Sc for the \(C(T;\mathbb {R}^{N})\)-norm topology.
The stronger conditions on A are the following:H(A)′: \(A\colon T\times \mathbb {R}^{N}\longrightarrow 2^{\mathbb {R}^{N}}\setminus \emptyset \) is a multifunction such that 0 ∈ A(t,0) for all t ∈ T, hypotheses H(A)′(i),(ii), and(iii) are the same as the corresponding hypotheses H(A)(i),(ii), and(iii) and
- (iv):
-
For every M > 0, there exists c > 0 such that
$$c_{M}|x-y|^{2}\leqslant (A(t,x)-A(t,y),x-y)_{\mathbb{R}^{N}} $$for all t ∈ T, all \(x,y\in \mathbb {R}^{N}\) with \(|x|,|y|\leqslant M\).
Example 6.2
Let \(\eta \in C(T;\mathbb {R}^{N})\), \(\eta \geqslant \widehat {c}>0\), \(A_{0}\colon \mathbb {R}^{N}\longrightarrow 2^{\mathbb {R}^{N}}\setminus \emptyset \) is a maximal monotone map with 0 ∈ A0(0) and set
with \(1<p\leqslant 2\). This map satisfies hypotheses H(A)′.
Theorem 6.3
If hypothesesH(A)′,H(φ) andH(F)4,H0,or \(H(F)_{4}^{\prime }\)holdwith
then, \(\overline {S}_{e}^{C(T;\mathbb {R}^{N})}=S_{c}\).
Proof
We follow the proof of Theorem 6.1, using this time instead of problem (6.7), the periodic problem (that is, the boundary condition in Eq. 6.7 will be u(0) = u(b)). So, \(\{u_{n}\}_{n\geqslant 1}\subseteq S_{e}\). Using the periodic boundary condition and hypothesis H(A)′(iv), we have (see Eq. 6.9 with t = b)
Passing to the limit as \(n\to +\infty \), we obtain
(see Eqs. 6.20 and 6.8), so \(\widehat {u}=u\) (see hypothesis H(A)′(iv)).
Therefore, \(u=\lim \limits _{n\rightarrow +\infty } u_{n}\) in \(C(T;\mathbb {R}^{N})\) with un ∈ Se for all \(n\in \mathbb {N}\) (see Eq. 6.8). □
7 An Example
In this section, we illustrate our results by examining the following periodic control system with a priori feedback:
In this problem,
Also, \(\mathbb {R}^{N}\ni x\longmapsto \partial |x|\) denotes the subdifferential in the sense of convex analysis. We know that
with \(\overline {B}_{1}=\{x\in \mathbb {R}^{N}:\ |x|\leqslant 1\}\) (see Gasiński and Papageorgiou [7]). The function \(f\colon T\times \mathbb {R}^{N}\longrightarrow \mathbb {R}^{N}\) is measurable in t ∈ T and locally L1(T)-Lipschitz in \(x\in \mathbb {R}\), that is, for every \(M^{\prime }>0\), there exists \(\eta ^{1}_{M^{\prime }}\in L^{1}(T)\) such that
The function \(v\colon T\longrightarrow \mathbb {R}^{m}\) is the control function and \(C\colon T\times \mathbb {R}^{N}\longrightarrow P_{kc}(\mathbb {R}^{m})\) is the control constraint multifunction. The dependence of C on \(x\in \mathbb {R}^{N}\) implies that there is a priori feedback in the system. We assume that
-
For all \(x\in \mathbb {R}^{N}\), t↦C(t, x) is graph measurable.
-
For almost all t ∈ T, x↦C(t, x) is locally L1(T) h-Lipschitz, that is, for every \(M^{\prime }>0\), there exists \(\eta _{M^{\prime }}^{2}\in L^{1}(T)\) such that
$$h(C(t,x),C(t,y))\leqslant \eta_{M^{\prime}}^{2}(t)|x-y|\quad\textrm{for a.a.}\ t\in T,\ \text{all}\ |x|,|y|\leqslant M^{\prime}. $$
Also, L is an N × m-matrix and we assume that there exist M > 0 and \(\widehat {a}_{M}\in L^{2}(T)\) such that
and
We set
This multifunction satisfies hypotheses H(F)4 while hypothesis H0 is clearly satisfied (see Eq. 7.2). Also, we have
Now, the control system (7.1) is equivalent to the following periodic differential inclusion:
We know that the solution set Sc of this multivalued system is nonempty (see Theorem 3.7) and in fact, we have
(see Proposition 3.8). Furthermore, if u ∈ Sc and ε > 0, then we can find \(u_{\varepsilon }\in W^{1,2}((0,b);\mathbb {R}^{N})\) such that
and \(\|u-u_{\varepsilon }\|_{C(T;\mathbb {R}^{N})}\leqslant \varepsilon \) (see Theorem 6.1).
In fact, if \(\xi (t)\geqslant \widehat {c}>0\) for all t ∈ T and \(1<p\leqslant 2\), then we can find \(u_{\varepsilon }\in W^{1,2}((0,b);\mathbb {R}^{N})\) satisfying (7.3) with periodic boundary condition (that is, uε(0) = uε(b)) and
(see Theorem 6.3).
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Acknowledgments
The authors wish to thank the three anonymous reviewers for their corrections and helpful remarks.
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The research was supported by the National Science Center of Poland under Project No. 2015/19/B/ST1/01169.
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Gasiński, L., Papageorgiou, N.S. Nonlinear Multivalued Periodic Systems. J Dyn Control Syst 25, 219–243 (2019). https://doi.org/10.1007/s10883-018-9408-9
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DOI: https://doi.org/10.1007/s10883-018-9408-9
Keywords
- Convex problem
- Nonconvex problem
- Extremal trajectories
- Strong relaxation
- Maximal monotone map
- Control system