Abstract
We study an optimal control problem with a nonlinear Volterra-type integral equation considered on a nonfixed time interval, subject to endpoint constraints of equality and inequality type, mixed state-control constraints of inequality and equality type, and pure state constraints of inequality type. The main assumption is the linear–positive independence of the gradients of active mixed constraints with respect to the control. We obtain first-order necessary optimality conditions for an extended weak minimum, the notion of which is a natural generalization of the notion of weak minimum with account of variations of the time. The conditions obtained generalize the corresponding ones for problems with ordinary differential equations.
Highlights
It is commonly known that, for problems with ordinary differential equations (ODEs), the theory of first order necessary optimality conditions including the Pontryagin maximum principle is completely developed
In [8], we considered a general problem with state and mixed constraints on a fixed time interval and obtained necessary conditions for the weak minimum
In Theorem 2 of this section we present the Lagrange multipliers rule (LMR) for a class of abstract nonsmooth optimization problems which contains Problem B as a special case
Summary
It is commonly known that, for problems with ordinary differential equations (ODEs), the theory of first order necessary optimality conditions including the Pontryagin maximum principle is completely developed It covers problems both on a fixed and a nonfixed time intervals containing pure state and mixed state-control constraints, as well as different types of integral and endpoint constraints. In the present paper we consider a general problem combining both a nonfixed time interval and state and mixed constraints, and obtain first order conditions for an extended weak minimum. We assume for simplicity that the function f is defined and twice continuously differentiable on an open set R ⊂ R2+n+r
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