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Abstract
In this paper, we derive some sufficient second order optimality conditions for control problems of partial differential equations (PDEs) when the cost functional does not involve the usual quadratic term for the control or higher nonlinearities for it. Though not always, in this situation the optimal control is typically bang-bang. Two different control problems are studied. The second differs from the first in the presence of the $L^1$ norm of the control. This term leads to optimal controls that are sparse and usually take only three different values (we call them bang-bang-bang controls). Though the proofs are detailed in the case of a semilinear elliptic state equation, the approach can be extended to parabolic control problems. Some hints are provided in the last section to extend the results.
Highlights
The aim of this paper is to prove some sufficient second order optimality conditions for optimal control problems of elliptic partial differential equations (PDEs) when the cost functional does not involve the control in an explicit form
The results that we present here cover the case of bang-bang controls
The main difference with the usual second order conditions is that the inequality J (u)v2 ≥ δ v 2 for every v in some cone of critical directions does not hold in general, and it has to be replaced for a weaker assumption, but one that is still strong enough to warrant the strict local optimality of the controls
Summary
The aim of this paper is to prove some sufficient second order optimality conditions for optimal control problems of elliptic partial differential equations (PDEs) when the cost functional does not involve the control in an explicit form. The analysis for control problems of ODEs is based on the assumption of a finite number of switching points, and at those points the derivative of the switching function does not vanish The extension of this approach to the case of PDEs is not clear at all. It is well known that the solution stability with respect to data perturbations and conditions for strict local optimality are closely related facts This justifies the attention paid to the second order analysis for control problems.
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