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Optimal control of linear retarded systems in Banach spaces. (English) Zbl 0603.49005
Consider the linear control problem with retarded arguments $dx(t)/dt=A_ 0x(t)+\int^{0}_{-h}d\eta (s)x(s+t)+f(t)+B(t)u(t)\quad a.e.\quad 0\leq t\leq T,$ $x(0)=g^ 0,\quad x(s)=g^ 1(s)\quad a.e.\quad s\in [-h,0).$ Here the state x(t) lies in a Banach space X, the control u(t) in a Banach space Y, $$A_ 0$$ generates a strongly continuous semigroup on X, and the Stieltjes measure $$\eta$$ is given by $\int^{0}_{-h}d\eta (s)x(s+t)=\sum^{m}_{r=1}A_ rx(t-h_ r)+\int^{0}_{-h}D(s)x(t+s)ds,$ where $$D(\cdot)\in L_{p'}([- h,0];{\mathcal L}(X))$$. The initial condition $$g=(g^ 0,g^ 1)$$ is given with $$g^ 0\in X$$, $$g^ 1\in L_ p([-h,0];X)$$. The author first studies the general linear system (without control), establishes existence and uniqueness of a mild solution, and derives a variation of constants formula for the mild solution in terms of the fundamental solution G(t), which is an operator in $${\mathcal L}(X)$$. Conditions are found under which a mild solution is a strong solution. The author now studies the optimal control problem with cost $J=\phi_ 0(x(T))+\int^{T}_{0}[f_ 0(x(t),t)+k_ 0(u(t),t)]dt$ where $$\phi_ 0:X\to R$$, $$f_ 0:X\times I\to R$$, $$k_ 0:Y\times I\to R$$, $$I=[0,T]$$, and where the admissible control set is closed and convex in $$L_ p(I;Y)$$. The adjoint system and fundamental solution are defined, and existence of an optimal control is proved under convexity hypotheses on $$\phi_ 0,f_ 0,k_ 0$$. Necessary conditions for optimality are obtained, characterized by the solution of the adjoint retarded system. A pointwise maximum principle is established for time-varying admissible control set and examples are given. The bang-bang principle is studied for the time optimal problem.
Reviewer: K.Cooke

##### MSC:
 49J27 Existence theories for problems in abstract spaces 49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 49K27 Optimality conditions for problems in abstract spaces 46B99 Normed linear spaces and Banach spaces; Banach lattices 93C05 Linear systems in control theory 93C25 Control/observation systems in abstract spaces 49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
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