Kazimierz Malanowski

Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems

A method of estimating the rate of convergence of approximation to convex, control-constrained optimal-control problems is proposed. In the method, conditions of optimality involving projections on the set of admissible control are exploited. General results are illustrated by examples of Galerkin-type approximations to optimal-control problems for parabolic systems.

Stability and sensitivity of solutions to nonlinear optimal control problems

Parameter-dependent optimal control problems for nonlinear ordinary differential equations, subject to control and state constraints, are considered. Sufficient conditions are formulated under which the solutions and the associated Lagrange multipliers are locally Lipschitz continuous and directionally differentiable functions of the parameter. The directional derivatives are characterized.

Sensitivity Analysis for Optimal Control Problems Subject to Higher Order State Constraints

A family of parameter dependent optimal control problems is considered. The problems are subject to higher-order inequality type state constraints. It is assumed that, at the reference value of the parameter, the solution exists and is regular. Regularity conditions are formulated under which the original problems are locally equivalent to some other problems subject to equality type constraints only. The classical implicit function theorem is applied to these new problems to investigate Fréchet dif ferentiability of the stationarity points with respect to the parameter.

The Lagrange-Newton method for nonlinear optimal control problems

We investigate local convergence of the Lagrange-Newton method for nonlinear optimal control problems subject to control constraints including the situation where the terminal state is fixed. Sufficient conditions for local quadratic convergence of the method based on stability results for the solutions of nonlinear control problems are discussed.

Second-order conditions and constraint qualifications in stability and sensitivity analysis of solutions to optimization problems in Hilbert spaces

The concepts of the strong second-order sufficient optimality condition and of the linear independence of gradients of active constraints play a crucial role in stability and sensitivity analysis of solutions to finite-dimensional mathematical programming problems. In this paper an attempt is made to use these concepts in stability and sensitivity analysis of solutions to cone-constrained optimization problems in Hilbert spaces. The abstract results are applied to optimal control problems for affine systems subject to state-space constraints.

Differentiability with respect to parameters of solutions to convex programming problems

We consider a family of convex programming problems that depend on a vector parameter, characterizing those values of parameters at which solutions and associated Lagrange multipliers are Gâteaux differentiable.These results are specialized to the problem of the metric projection onto a convex set. At those points where the projection mapping is not differentiable the form of Clarkes generalized derivative of this mapping is derived.

Sensitivity Analysis of Optimization Problems in Hilbert Space with Application to Optimal Control

A family of optimization problems in a Hilbert space depending on a vector parameter is considered. It is assumed that the problems have locally isolated local solutions. Both these solutions and the associated Lagrange multipliers are assumed to be locally Lipschitz continuous functions of the parameter. Moreover, the assumption of the type of strong second-order sufficient condition is satisfied.It is shown that the solutions are directionally differentiable functions of the parameter and the directional derivative is characterized. A second-order expansion of the optimal-value function is obtained. The abstract results are applied to state and control constrained optimal control problems for systems described by nonlinear ordinary differential equations with the control appearing linearly.

Differential stability of solutions to convex, control constrained optimal control problems

A family of convex, control constrained optimal control problems that depend on a real parameter is considered. It is shown that under some regularity conditions on data the solutions of these problems, as well as the associated Lagrange multipliers are directionally differentiable with respect to parameter. The respective right-derivatives are given as the solution and the associated Lagrange multipliers for some quadratic optimal control problem. If a condition of strict complementarity type hold, then directional derivatives become continuous ones.

The Lagrange-Newton method for state constrained optimal control problems

Local convergence of the Lagrange-Newton method for optimization problems with two-norm discrepancy in abstract Banach spaces is investigated. Based on stability analysis of optimization problems with two-norm discrepancy, sufficient conditions for local superlinear convergence are derived. The abstract results are applied to optimal control problems for nonlinear ordinary differential equations subject to control and state constraints.

Stability and Sensitivity of Solutions to Optimal Control Problems for Systems with Control Appearing Linearly

We consider a family of optimal control problems for systems described by nonlinear ordinary differential equations with control appearing linearly. The cost functionals and the control constraints are convex. All data depend on a vector parameter.Using the concept of the second-order sufficient optimality conditions it is shown that the solutions of the problems, as well as the associated Lagrange multipliers, are locally Lipschitz continuous and directionally differentiable functions of the parameter.