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Control problems governed by time-dependent maximal monotone operators. (English) Zbl 1367.34083

The authors first consider the evolution inclusion \[ \begin{cases}-\dot{x}(t)\in A(t)x(t)+f(t,x(t))\text{ a.e. on }I\equiv [ 0,T], \\ x(0)=a\in D\left( A(0)\right)\end{cases} \] in a real Hilbert space \(H\), where \(A(t):D\left( A(t\right) )\rightarrow 2^{H}\) is a maximal monotone operator for each \(t\in I\), and the perturbation \(f\) maps \(I\times H\) into \(H\). Using methods from M. Kunze and M. D. P. Monteiro Marques [Set-Valued Anal. 5, No. 1, 57–72 (1997; Zbl 0880.34017)] and the first author et al. [Numer. Funct. Anal. Optim. 34, No. 10, 1156–1186 (2013; Zbl 1288.34057)], the existence of a unique absolutely continuous solution is proven under the assumptions that \(A\) satisfies a continuity-type property and a growth condition, \(f\) is separately measurable, Lipschitz continuous in its second variable on bounded subsets of \(H\) and also satisfies a growth condition. Estimates for the solution are also derived, and the map \(\psi :a\rightarrow x_{a}\) is shown to be Lipschitz continuous.
In the second part of the article, under the additional assumption that \(H\) is separable, the existence and uniqueness result above is used to study a Bolza type optimal control problem associated with such an evolution inclusion using Young measures. More specifically, it is shown that the control problem \[ \inf_{\varsigma }\int\nolimits_{0}^{T}J(t,x_{\varsigma }(t),\varsigma (t))dt, \] has an optimal solution that is given by the relaxed problem \[ \inf_{\mu }\int\nolimits_{0}^{T}\int\nolimits_{U}J(t,x_{\mu }(t),u)\mu _{t}(du)dt \] where \(U\) is a compact metric space, \(J\) is bounded below, satisfies an integrability-type condition, and for each \(t\in I\), \(J(t,\cdot ,\cdot )\) is continuous. \(x_{\varsigma }\) is the solution to \[ \begin{cases} -\dot{x}(t)\in A(t)x(t)+g(t,x(t),\varsigma (t))\text{ a.e. on }I, \\ x(0)=x_{0},\end{cases} \] where \(g\) satisfies appropriate conditions, and \(x_{\mu }\) is the solution to the problem \[ \begin{cases} -\dot{x}(t)\in A(t)x(t)+\int\nolimits_{\Gamma (t)}g(t,x(t),u)\mu _{t}(du)\text{ a.e. on }I, \\ x(0)=x_{0},\end{cases} \] where \(\Gamma :I\rightarrow 2^{U}\) is a compact-valued and measurable. The approach is similar to the one from J. F. Edmond and L. Thibault [Math. Program. 104, No. 2–3 (B), 347–373 (2005; Zbl 1124.49010)].

MSC:

34G25 Evolution inclusions
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34H05 Control problems involving ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
49K27 Optimality conditions for problems in abstract spaces
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References:

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