Operator theoretic approach to optimal control problems described by nonlinear differential equations

Abstract

We consider a class of control systems characterized by nonlinear differential equation of the form dx/dx = A(t)x + B(t)u + F(t,x),0≤t o≤t≤t1<∞. x(to) = xo where u denotes the control lying in a suitable Banach space and x denotes the state in another separable reflexive Banach space. We are interested in finding a control, u which minimizes a certain cost functional J(u) = ø(x,u). We provide conditions on A(t),B(t),F(t,x) and ø(t,u) which gurantee the existence of an optimal control. We first reduce the system governed by the differential equation into an equivalent Hammerstein operator equation of the form x = KNx + Hu in suitable space. Subsequently we give sets of sufficient conditions on operators K, N and H which guarantee the existence of an optimal control. We use the theory of monotone operators and operators of type (M) in our analysis. Our results apply to both Lipschitzian and non-Lipschitzian (monotone) nonlinearities. The systems described by standard finite and infinite dimensional nonlinear differential equations are special cases of the general operator equation formulation. From the general results obtained for the operator equation we deduce results for the system described by differential equations as special cases. Also, we relate 'optimality system' to Hamiltonion system in the Minimum Principle of Pontriagin and Riccati Equations for systems governed by differential equations.