# American Institute of Mathematical Sciences

2016, 6(3): 221-239. doi: 10.3934/naco.2016009

## Partial stabilizability and hidden convexity of indefinite LQ problem

 1 Dept. Systems Engineering, University of Valladolid, 47005 Valladolid, Spain 2 Dept. Systems Engineering, Research School Of Information Sciences And Engineering, The Australian National University, Canberra, Act 0200, Australia

Received  August 2015 Revised  July 2016 Published  September 2016

Generalization of linear system stability theory and LQ control theory are presented. It is shown that the partial stabilizability problem is equivalent to a Linear Matrix Inequality (LMI). Also, the set of all initial conditions for which the system is stabilizable by an open-loop control (the stabilizability subspace) is characterized in terms of a semi-definite programming (SDP). Next, we give a complete theory for an infinite-time horizon Linear Quadratic (LQ) problem with possibly indefinite weighting matrices for the state and control. Necessary and sufficient convex conditions are given for well-posedness as well as attainability of the proposed (LQ) problem. There is no prior assumption of complete stabilizability condition as well as no assumption on the quadratic cost. A generalized algebraic Riccati equation is introduced and it is shown that it provides all possible optimal controls. Moreover, we show that the solvability of the proposed indefinite LQ problem is equivalent to the solvability of a specific SDP problem.
Citation: Mustapha Ait Rami, John Moore. Partial stabilizability and hidden convexity of indefinite LQ problem. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 221-239. doi: 10.3934/naco.2016009
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