In this paper we study optimal control problems governed by semilinear parabolic equations. We obtain necessary optimality conditions in the form of an exact Pontryagin's minimum principle for distributed and boundary controls (which can be unbounded) and bounded initial controls. These optimality conditions are obtained thanks to new regularity results for linear and nonlinear parabolic equations.

Hamiltonian Pontryagin's Principles for Control Problems Governed by Semilinear Parabolic Equations

This paper deals with optimal control problems governed by semilinear parabolic equations with pointwise state constraints and unbounded controls. Under some strong stability assumption, we obtain necessary optimality conditions in the form of a Pontryagin's minimum principle in qualified form. A Pontryagin's principle in nonqualified form is also proved without any stability condition.

Pontryagin's Principle for State-Constrained Control Problems Governed by Parabolic Equations with Unbounded Controls

This paper is concerned with the state constrained optimal control problems of a fractional diffusion equation in a bounded domain. The fractional time derivative is considered in the Riemann-Liouville sense. Under a Slater type condition we prove the existence a Lagrange multiplier and a decoupled optimality system.

Optimal control of a fractional diffusion equation with state constraints

We present variants of a convergent Lagrangean relaxation algorithm for minimizing a strictly convex separable quadratic function over a transportation polytope. The algorithm alternately solves two “subproblems,” each of which has an objective function that is defined by using Lagrange multipliers derived from the other. Motivated by the natural separation of the subproblems into independent and very easily solved “subsubproblems,” the algorithm can be interpreted as the cyclic coordinate ascent method applied to the dual problem. We exhibit our computational results for different implementations of the algorithm applied to a set of large constrained matrix problems.

A Lagrangean relaxation algorithm for the constrained matrix problem

In this paper we consider the problem of finding the constrained global optimum of an indefinite quadratic function. Since such a function may have many local optima, finding the global optimum is a computationally difficult problem. We give an overview of the most important methods used, which include Benders decomposition, concave programming approaches, enumerative techniques, and bilinear programming.

Global optimization algorithms for linearly constrained indefinite quadratic problems

Convex parabolic boundary control problems with pointwise state constraints

Dualität und Approximation bei konvexen Optimierungsproblemen

Convex optimal control problems for parabolic distributed parameter systems with an integral constraint for the gradient of the state are considered. The objective function is a general convex integral functional in which the control constraints are incorporated. Necessary and sufficient conditions for optimality are derived.

Uwe Mackenroth

Papers

Convex parabolic boundary control problems with pointwise state constraints

Dualität und Approximation bei konvexen Optimierungsproblemen

On parabolic distributed optimal control problems with restrictions on the gradient

Weak duality for parabolic boundary control problems

Some general considerations on optimality conditions for state constrained parabolic control problems