Minimum In Optimal Control Problems Research Articles

A second-order sufficient condition for a weak local minimum in an optimal control problem with an inequality control constraint

Abstract This paper is devoted to a sufficient second-order condition for a weak local minimum in a simple optimal control problem with one control constraint G(u) ≤ 0, given by a C 2-function. A similar second-order condition was obtained earlier by the author for a strong minimum in a much more general problem. In the present paper, we would like to take a narrower perspective than before and thus provide shorter and simpler proofs. In addition, the paper uses the first and second order tangents to the set U, defined by the inequality G(u) ≤ 0. The main difficulty of the proof, clearly shown in the paper, refers to the set, where the gradient H u of the Hamiltonian is small, but the condition of quadratic growth of the Hamiltonian is satisfied. The paper can be valuable for self-explanation and provides a basis for extensions.

Necessary second-order conditions for a weak local minimum in a problem with endpoint and control constraints

One of the first steps towards necessary second-order optimality conditions in problems with constraints was taken by Dubovitskii and Milyutin in 1965. They offered a scheme that was very effective in smooth optimization problems, but seemed to be not suitable for applications in problems with pointwise control constraints. In this article we consider a modification of the Dubovitskii–Milyutin scheme, which allows to derive necessary second-order conditions for a weak local minimum in optimal control problems with a finite number of endpoint constraints of equality and inequality type and with pointwise control constraints of inequality type given by smooth functions. Assuming that the gradients of active control constraints are linearly independent, we provide rather straightforward proof of these conditions for a measurable and essentially bounded optimal control.

Necessary Conditions for a Weak Minimum in Optimal Control Problems with Integral Equations Subject to State and Mixed Constraints

The first order necessary optimality conditions for a weak minimum are derived for optimal control problems with Volterra-type integral equations, considered on a fixed 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 uniform linear-positive independence of the gradients of active mixed constraints with respect to the control. The conditions obtained generalize the Euler--Lagrange equation (as a stationarity condition) for the Lagrange problem in the classical calculus of variations with ordinary differential equations.

Singular Regimes in Certain Classes of Relaxed Control Problems

We present new necessary conditions for a relaxed minimum in optimal control problems defined by certain classes of ordinary differential equations. These necessary conditions may be helpful in computing singular extremal arcs, in determining when these arcs are “strictly relaxed”, and in defining regions of the state space that contain nonsingular extremal arcs only.

On the conjugate point condition for the control problem

In this paper the conjugate point condition in the classical calculus of variations is derived from a control-theoretic point of view. The relationship of conjugate point condition to relative minimum in optimal control problem is discussed.