Optimal control of linear retarded systems in Banach spaces.

*(English)*Zbl 0603.49005Consider 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.) |

##### Keywords:

linear control problem with retarded arguments; Banach space; mild solution; adjoint system; pointwise maximum principle; bang-bang principle; time optimal problem
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\textit{S.-i. Nakagiri}, J. Math. Anal. Appl. 120, 169--210 (1986; Zbl 0603.49005)

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