TY - JOUR

T1 - Partial stabilizability and hidden convexity of indefinite LQ problem

AU - Rami, Mustapha Ait

AU - Moore, John

N1 - Publisher Copyright:
© 2016 American Institute of Mathematical Sciences. All rights reserved.

PY - 2016/9

Y1 - 2016/9

N2 - 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 con-ditions 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 as-sumption on the quadratic cost. A generalized algebraic Riccati equation is introduced and it is shown that it provides all possible optimal controls. More-over, we show that the solvability of the proposed indefinite LQ problem is equivalent to the solvability of a specific SDP problem.

AB - 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 con-ditions 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 as-sumption on the quadratic cost. A generalized algebraic Riccati equation is introduced and it is shown that it provides all possible optimal controls. More-over, we show that the solvability of the proposed indefinite LQ problem is equivalent to the solvability of a specific SDP problem.

KW - Generalized algebraic riccati equation

KW - Indefinite LQ problem

KW - Linear matrix inequality

KW - Semi-definite programming

KW - X-stabilizability

UR - http://www.scopus.com/inward/record.url?scp=84989318890&partnerID=8YFLogxK

U2 - 10.3934/naco.2016009

DO - 10.3934/naco.2016009

M3 - Article

SN - 2155-3289

VL - 6

SP - 221

EP - 239

JO - Numerical Algebra, Control and Optimization

JF - Numerical Algebra, Control and Optimization

IS - 3

ER -