Optimal control of nonlinear evolution equations (Q2746643)

Most of the results of this very large and consistent article concern minimax optimal control problems of the form: NEWLINE\[NEWLINE \beta := \inf_{u(.)} \sup_{\lambda(.)}J(u(.),\lambda(.))= \sup_{\lambda(.)}J(u^*(.),\lambda(.)) NEWLINE\]NEWLINE subject to: NEWLINE\[NEWLINE J(u(.),\lambda(.)):=\int_a^bL(t,x(t),u(t)) dt NEWLINE\]NEWLINE NEWLINE\[NEWLINE x'(t)+A(t,x(t))=\int_Vf(t,x(t),v)\lambda(t)(dv)+B(t)u(t), \quad x(a)=x_0\in H,NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(t)\in U(t), \quad \lambda(t)\in\Sigma(t) \text{ a.e. on }T=[a,b], \quad u(.)\in L^q(T,Y) NEWLINE\]NEWLINE in a rather complicated measure-theoretical infinite dimensional setting. In Section 3 the authors prove several auxiliary results and a theorem stating the existence of an optimal control, \(u^*(.)\), and in Section 5 they obtain in Theorem 5 certain necessary conditions for a saddle point \((u^*(.), \lambda^*(.))\) of the problem. NEWLINENEWLINENEWLINEIn Section 4 the authors prove the existence of optimal controls for both, minimal-time and Bolza optimal control problems associated to evolution inclusions of the form: NEWLINE\[NEWLINE x'(t)+A(t,x(t))\in F(t,x(t)), \quad x(0)=x_0\in H,NEWLINE\]NEWLINE in a similar setting, while in the last section the abstract results are applied to three examples of nonlinear parabolic distributed control systems.