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A general statement of dual first-order sufficient optimality conditions for the generalized problem of Bolza

Galewska E.

Demonstratio Mathematica

2003

Vol. 36, nr 3

711--720

In this paper we provide first-order sufficient optimality conditions for the generalized problem of Bolza when all arcs take values in a separable Hilbert space. Our approach consists in the explicit construction of a quadratic function that satisfies the dual Hamilton-Jacobi inequality. The essential role in the generalized conditions plays the existence of a certain function for which a certain inequality holds.

Control problems for systems described by hemivariational inequalities

Denkowski Z.

Control and Cybernetics

2002

Vol. 31, no 3

713-738

The paper considers some control problems for the systems described by the evolution, as well as the stationary hemivariational inequalities (HVIs for short). First, basing on surjectivity theorems for pseudo-monotone operators we formulate some existence results for the solutions of the HVIs and investigate some properties of the solution set (like sensitivity; i.e. its dependence on data and operators). Next we quote some existence theorems for optimal solutions for various classes of optimal control like distributed control (e.g. Bolza problem), identification of parameters, or optimal shape design for systems described by HVIs. Finally, we discuss some common features in getting the existence of optimal solutions as well as some "well-posedness" problems.

Theconjugate points sufficient conditions for an optimal control

Szymanowska A.

Control and Cybernetics

1998

Vol. 27, no 3

353-362

The paper concerns an application of the idea of field theory and the concept of "concourse of flights" to the sufficient optimality conditions for the optimal control problems stated in terms of focal and conjugate points. The concept of concourse of flights was begun by Young (1969), and later extended by Nowakowski (1988). In the paper the definition of a focal and conjugate point of a field of extremals is given. Using these concepts, we prove that the existence of a field of extremals without conjugate points implies the existence of concourse of flights and consequently we obtain the second order sufficient conditions for the generalized problem of Bolza. Another approach to the concept of focal and conjugate points is given by Zeidan (1983, 1984).