Optimal control of nonlinear second order evolution equations. (English) Zbl 0818.49001
The authors consider an optimal control problem for a class of nonlinear second order infinite-dimensional systems in the setting of the Gelfand triple \(X\hookrightarrow H\hookrightarrow X^*\) with injection being continuous and compact. The problem they consider can be formulated as follows:
\[
\begin{cases} J(u, x)= \int_ I L(t, x, \dot x, u)dt\to\inf\\ \text{subject to}:\\ \ddot x+ A(t, \dot x)+ Bx= f(t, x)u,\;x(0)= x_ 0\in X,\;\dot x(0)= x_ 1\in H,\;u(t)\in U(t)\text{ a.e.},\end{cases}
\]
where \(U\) is a measurable set-valued map with values in \(2^ Y\), \(Y\) being a separable reflexive Banach space, and \(A\) is a nonlinear operator from \(I\times X\to X^*\), \(B\in {\mathcal L}(X, X^*)\), \(f: I\times H\to {\mathcal L}(Y,H)\).
Under some suitable assumptions the authors prove the existence of optimal controls for the problem as stated. This result is an extension of an earlier work of N. S. Papageorgiou [Glas. Mat., III. Ser. 27, No. 2, 297-311 (1992; Zbl 0806.49009)].
Under some suitable assumptions the authors prove the existence of optimal controls for the problem as stated. This result is an extension of an earlier work of N. S. Papageorgiou [Glas. Mat., III. Ser. 27, No. 2, 297-311 (1992; Zbl 0806.49009)].
Reviewer: N.U.Ahmed (Ottawa)
MSC:
49J20 | Existence theories for optimal control problems involving partial differential equations |
49J27 | Existence theories for problems in abstract spaces |
34G20 | Nonlinear differential equations in abstract spaces |
34H05 | Control problems involving ordinary differential equations |