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Kalman duality principle for a class of ill-posed minimax control problems with linear differential-algebraic constraints. (English) Zbl 1287.49026

Summary: In this paper we present a Kalman duality principle for a class of linear Differential-Algebraic Equations (DAE) with arbitrary index and time-varying coefficients. We apply it to an ill-posed minimax control problem with DAE constraint and derive a corresponding dual control problem. It turns out that the dual problem is ill-posed as well and so classical optimality conditions are not applicable in the general case. We construct a minimizing sequence \(\hat{u}_{\varepsilon}\) for the dual problem applying Tikhonov’s method. Finally, we represent \(\hat{u}_{\varepsilon}\) in the feedback form using the Riccati equation on a subspace which corresponds to the differential part of the DAE.

MSC:

49K35 Optimality conditions for minimax problems
49N15 Duality theory (optimization)
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