# Kalman Duality Principle for a Class of Ill-Posed Minimax Control Problems with Linear Differential-Algebraic Constraints

Kalman Duality Principle for a Class of Ill-Posed Minimax Control Problems with Linear... In this paper we present 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 method. Finally we represent $\hat{u}_{\varepsilon}$ in the feedback form using Riccati equation on a subspace which corresponds to the differential part of the DAE. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Kalman Duality Principle for a Class of Ill-Posed Minimax Control Problems with Linear Differential-Algebraic Constraints

, Volume 68 (2) – Oct 1, 2013
21 pages

/lp/springer_journal/kalman-duality-principle-for-a-class-of-ill-posed-minimax-control-03Ejfi3axr
Publisher
Springer US
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-013-9207-3
Publisher site
See Article on Publisher Site

### Abstract

In this paper we present 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 method. Finally we represent $\hat{u}_{\varepsilon}$ in the feedback form using Riccati equation on a subspace which corresponds to the differential part of the DAE.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Oct 1, 2013

### References

• Minimax state estimation for linear discrete-time differential-algebraic equations
Zhuk, S.

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