A nonquadratic Bolza problem and a quasi-Riccati equation for distributed parameter systems. (English) Zbl 0632.49004
An optimal control problem for a linear evolution equation in a Hilbert space X is considered, \(dx/dt=Ax(t)+Bu\), \(x(0)=x_ 0\) where A is the infinitesimal generator of a \(C_ 0\) semigroup, \(B\in {\mathcal L}(V;X)\), V is the Hilbert space of controls. The cost function includes final and distributed in time observations
\[
J(u)=M(x(T))+\int^{T}_{0}(Q(x(t))+(1/2)<Ru(t),\quad u(t)>)dt
\]
where M and Q are \(C^ 2\) convex functionals.
First the existence and uniqueness of an optimal control is proved and the first order necessary optimality conditions are given in the form of an integral equation. Then a quasi-Riccati operator equation is introduced. A normal solution of that equation is a nonlinear operator P(t,x) such that (1) P(t,\(\cdot)\) is a gradient operator (2) the corresponding closed-loop state equation has a global mild solution, and (3) having certain regularities. Any such normal solution yields a closed-loop optimal control. Then the nonlinear integral equation giving the optimal adjoint state at time s knowing the state at time t (t\(\leq s)\) is studied. Its solution mapping is denoted by K(s,t,x). Results on the existence, uniqueness and regularity of K are given. Finally, it is shown that \(P(t,x)=K(t,t,x)\) is a normal solution of the quasi-Riccati equation. An extension to differential games is proposed.
First the existence and uniqueness of an optimal control is proved and the first order necessary optimality conditions are given in the form of an integral equation. Then a quasi-Riccati operator equation is introduced. A normal solution of that equation is a nonlinear operator P(t,x) such that (1) P(t,\(\cdot)\) is a gradient operator (2) the corresponding closed-loop state equation has a global mild solution, and (3) having certain regularities. Any such normal solution yields a closed-loop optimal control. Then the nonlinear integral equation giving the optimal adjoint state at time s knowing the state at time t (t\(\leq s)\) is studied. Its solution mapping is denoted by K(s,t,x). Results on the existence, uniqueness and regularity of K are given. Finally, it is shown that \(P(t,x)=K(t,t,x)\) is a normal solution of the quasi-Riccati equation. An extension to differential games is proposed.
Reviewer: J.Henry
MSC:
49J27 | Existence theories for problems in abstract spaces |
47A62 | Equations involving linear operators, with operator unknowns |
49K27 | Optimality conditions for problems in abstract spaces |
93C20 | Control/observation systems governed by partial differential equations |
35B37 | PDE in connection with control problems (MSC2000) |
35G10 | Initial value problems for linear higher-order PDEs |
35K25 | Higher-order parabolic equations |
47D03 | Groups and semigroups of linear operators |
49J20 | Existence theories for optimal control problems involving partial differential equations |
49K20 | Optimality conditions for problems involving partial differential equations |
93C25 | Control/observation systems in abstract spaces |