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Least regret control, virtual control and decomposition methods. (English) Zbl 0951.65060

The author introduces a “least regret control” on a simple example. Let \(A\) be a second-order elliptic operator and the state \(z\) be given by the solution of the system: \[ Az=v1_{\mathcal O},\quad \text{in } \Omega, \qquad \frac{\partial z}{\partial n_A}=g\quad \text{on} \Gamma, \] where \(1_{\mathcal O}\) is the characteristic function of \({\mathcal O}\subset \Omega\), \(v\) is the control, and \(g\) is a perturbation (or unknown) variable. \(J(v,g)\) is a quadratic functional. The solution of the following problem \[ \inf_{v} \sup_{g} \biggl\{J(v,g)-J(v_0, g)-\gamma/2 \int_{\Gamma} g^2 d\Gamma \biggr\} \] is called the “least regret control”. By use of augmented state equations the above problem is reduced to a standard optimal control one. Moreover, the author introduces the method of virtual control, aimed at the “decomposition of everything” (decomposition of the domain, of the operator, etc.) and presents an algorithm. Finally, he introduces many references on the above methods and the remarks.

MSC:

65K10 Numerical optimization and variational techniques
49J20 Existence theories for optimal control problems involving partial differential equations
49M27 Decomposition methods
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References:

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