Volume 2, Number 3 SYSTEMS & CONTROL LETTERS October 1982 Invariance of the quadratic cost non-optimal control laws on a set of M.M. KONSTANTINOV Institute of Engineering Cybernetics & Robotics, A kad, G. Boncheu Str., BI. 12," 1113 Sofia, Bulgaria P.Hr. PETKOV, N.D. CHRISTOV Department of Automatics, Higher Institute of Mechanical & Electrical Engineering," 1156 Sofia, Bulgaria Received 8 May 1982 A linear time-invariant system with a quadratic cost (q.c.) is studied and conditions are found for the q.c. to be invariant under the action of various non-optimal control laws. The set of stabilizing control laws is divided into equivalence classes (the q.c. being constant on each of them) and the complete description of the quotient space is given. A number of related problems is discussed and illustrative examples are considered. Keywords: Linear time-invariant systems, Linear-quadratic optimization, Inverse optimal control problem. 1. Introduction Given a standard linear-quadratic optimization problem, two phenomena are worth mentioning: (i) There exists a subspace of initial states which generate the optimal input and state trajectories provided a non-optimal (and even a non-stabilizing) control law is applied [1], [2]. (ii) The quadratic cost (q.c.) is invariant on a set of non-optimal trajectories [1]. In connection with (ii) the concept of equivalence on the set of stabilizing control laws is introduced in this paper (the equivalent control laws produce the same value of the q.c.), and the structure of the corresponding quotient space is studied. The following abbreviations are used later: R ..... - the space of real n × m-matrices (R" 1 -= R n); A'r _ the transpose of A; S(n)C R "'~ (S+(n)C S(n)) - the set of symmetric (positive semidefinite) matrices; Si+ (n) - the set of all P E S+(n) of rank less or equal to i (S~ + (n) = S+(n)); O(m) C R .... - the group of orthogonal matrices; SO(m) C O(m) - the group of orthogonal matrices with determinant equal to 1. 2. Problem statement Consider the stabilizable and detectable time-invariant system =Ax(t) +Bu(t), x(0)=x0, (1) y(t) = Cx(t), where x(t) E R", u(t) ~ R m, y(t) E g~r. Denote by D = (K: the matrix Ac(K ) =A + BKis stable) C n"" the set of stabilizing gain matrices, and let the class of admissible time-invariant control laws be u(t)----Kx(t), KED. (2) 0167-691 i/82/0000-0000/$02.75 © 1982 North-Holland 169