Nonlinear Multivalued Periodic Systems
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Abstract
We consider a firstorder periodic system involving a timedependent 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.
Keywords
Convex problem Nonconvex problem Extremal trajectories Strong relaxation Maximal monotone map Control systemMathematics Subject Classification (2010)
34A34 34B15 47N201 Introduction
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 timeindependent with \(D(A)\ne \mathbb {R}^{N}\). The author proves existence theorems for both the convex and nonconvex problems using the notion of L^{ p }solution tube. Qin and Xue [14] assume that that A is timeindependent and is a positive definite N × Nmatrix. 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 HuPapageorgiou [11]) and from the theory of nonlinear operators of monotone type (see GasińskiPapageorgiou [7] and Zeidler [15]).
Since χ_{ω∖C} = 1 − χ_{ C }, 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 →2^{ Z } ∖∅ 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, d_{ Z } denotes the metric of Z).
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 thatfor all \(0\leqslant s\leqslant t\leqslant b\) and all (u_{1}, h_{1}) ∈Gr A(t,⋅), (u_{2}, h_{2}) ∈Gr A(s,⋅).$$(h_{1}h_{2},u_{1}u_{2})_{\mathbb{R}^{N}}\geqslant \eta(t)\eta(s)u_{1}u_{2}l(\max\{u_{1},u_{2}\}) $$
 (iii)
 For every r > 0, there exists a_{ r } ∈ L^{2}(T) such thatand for all \(u\in L^{2}(T;\mathbb {R}^{N})\), t↦A^{0}(t, u(t)) belongs in \(L^{2}(T;\mathbb {R}^{N})\).$$A(t,x)\leqslant a_{r}(t)\quad\textrm{for a.a.}\ t\in T,\ \text{all}\ x\leqslant r $$
Remark 2.1
Remark 2.2
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]).
Lemma 2.3
If \(A\colon X\longrightarrow 2^{X^{*}}\) is a maximal monotone map with 0 ∈ A(0), then \(\mathcal {A}\colon L^{p}(T;X)\longrightarrow 2^{L^{p^{\prime }}(T;X^{*})}\) is maximal monotone.
Proof
 Claim

\(R(\mathcal {A}+\vartheta )=L^{p^{\prime }}(T;X^{*})\) (that is, \(\mathcal {A}+\vartheta \) is surjective).
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
Together with H(F)_{1}, we will need the following extra condition on ∂φ:
 H_{0}:

For all x ∈ D(∂φ) and g ∈ ∂φ(x), we have \((g,x)_{\mathbb {R}^{N}}\geqslant 0\).
Alternatively, instead of H(F)_{1}, H_{0}, 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 ∈ L^{2}(T).
Proposition 3.2
If hypotheses H(A) and H(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
Proposition 3.3
If hypotheses H(A) and H(φ) hold, then the Poincaré map P is a contraction.
Proof
Proposition 3.4
Proof
Proposition 3.5
 (a)

If hypotheses H(A), H(φ), H(F)_{1}, and H_{0} hold, then \(u(t)\leqslant M\) for all t ∈ T, all \(u\in \widehat {S}_{\varepsilon }\)(here, M > 0 is as postulated in hypothesis H(F)_{1}(iii)).
 (b)

If hypotheses H(A), H(φ), and \(H(F)_{1}^{\prime }\) hold, then there exists M > 0 such that \(u(t)\leqslant M\) for all t ∈ T, all u ∈ S_{ ε }.
Proof
 (I)

u(t) > M for all t ∈ T.
 (II)
 There exist \(0\leqslant \eta \leqslant \tau \leqslant b\) such that$$u(\eta)=M\ \text{and} \ u(t)>M \quad\forall t\in [\eta,\tau]. $$
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 hypotheses H(A) and H(φ) hold, then the solution map \(\gamma _{\varepsilon }\colon L^{2}(T;\mathbb {R}^{N})\longrightarrow C(T;\mathbb {R}^{N})\) is completely continuous (that is, if \(h_{n} \overset {w}{\longrightarrow } h\) in \(L^{2}(T;\mathbb {R}^{N})\), then γ_{ ε }(h_{ n })→γ_{ ε }(h) in \(C(T;\mathbb {R}^{N})\)).
Proof
Now, we can have our existence theorem for the “convex problem.”
Theorem 3.7
If hypotheses H(A), H(φ) and H(F)_{1}, H_{0} or \(H(F)_{1}^{\prime }\) hold, then problem (1.1) admits a solution \(u_{0}\in W^{1,2}((0,b);\mathbb {R}^{N})\).
Proof
The set W furnished with the relative weak topology is compact. Also, it is convex. Then, invoking the KakutaniKy 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 \(g_{0}^{\varepsilon }\in S_{F(\cdot ,\gamma _{\varepsilon }(g_{0}^{\varepsilon })(\cdot ))}^{2}\).
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 hypotheses H(A), H(φ) and H(F)_{1}, H_{0}, 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 H_{0}. Alternatively, instead of the pair H(F)_{2}, H_{0}, 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 ∈ L^{2}(T).
Theorem 4.1
If hypotheses H(A), H(φ) and H(F)_{2}, H_{0}, 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 ArzelaAscoli theorem implies that the sequence \(\{u_{0}^{\varepsilon _{n}}\}_{n\geqslant 1}\subseteq C(T;\mathbb {R}^{N})\) is relatively compact.
5 Extremal Trajectories
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 hcontinuous.
 (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 H_{0}. Alternatively, we can replace the pair H(F)_{3}, H_{0} 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 ∈ L^{2}(T).
Theorem 5.1
If hypotheses H(A), H(φ) and H(F)_{3}, H_{0}, 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
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 \(u_{0}^{\varepsilon }=\sigma _{\varepsilon }(u_{0}^{\varepsilon })\).
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 “bangbang 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 η_{ r } ∈ L^{1}(T) such thatfor almost all t ∈ T, all \(x,y\in \mathbb {R}^{N}\) with \(x,y\leqslant r\).$$h(F(t,x),F(t,y))\leqslant \eta_{r}(t)xy $$
 (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 H_{0}. Alternatively, the pair H(F)_{4}, H_{0} 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 ∈ L^{2}(T).
In what follows by S_{ c }, 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 \(S_{e}(x_{0})\subseteq W^{1,p}((0,b);\mathbb {R}^{N})\) be the solution set of Eq. 6.1. A simplified version of the proof of Theorem 5.1 shows that S_{ e }(x_{0})≠∅. Then, our strong relaxation result reads as follows.
Theorem 6.1
If hypotheses H(A), H(φ) and H(F)_{4}, H_{0}, or \(H(F)_{4}^{\prime }\) hold and u ∈ S_{ c }, then there exists a sequence \(\{u_{n}\}_{n\geqslant 1}\subseteq S_{e}(u(0))\) such that u_{ n }→u in \(C(T;\mathbb {R}^{N})\).
Proof
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 S_{ e }≠∅. If we strengthen the conditions on A(t,⋅), we can show that the set S_{ e } is dense in S_{ c } 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 thatfor all t ∈ T, all \(x,y\in \mathbb {R}^{N}\) with \(x,y\leqslant M\).$$c_{M}xy^{2}\leqslant (A(t,x)A(t,y),xy)_{\mathbb{R}^{N}} $$
Example 6.2
Theorem 6.3
Proof
Therefore, \(u=\lim \limits _{n\rightarrow +\infty } u_{n}\) in \(C(T;\mathbb {R}^{N})\) with u_{ n } ∈ S_{ e } for all \(n\in \mathbb {N}\) (see Eq. 6.8). □
7 An Example

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 L^{1}(T) hLipschitz, that is, for every \(M^{\prime }>0\), there exists \(\eta _{M^{\prime }}^{2}\in L^{1}(T)\) such that$$h(C(t,x),C(t,y))\leqslant \eta_{M^{\prime}}^{2}(t)xy\quad\textrm{for a.a.}\ t\in T,\ \text{all}\ x,y\leqslant M^{\prime}. $$
Notes
Acknowledgments
The authors wish to thank the three anonymous reviewers for their corrections and helpful remarks.
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