794 ISSN 1064–5624, Doklady Mathematics, 2007, Vol. 76, No. 2, pp. 794–796. © Pleiades Publishing, Ltd., 2007. Original Russian Text © D.V. Balandin, M.M. Kogan, 2007, published in Doklady Akademii Nauk, 2007, Vol. 416, No. 5, pp. 606–609. The classical optimal linear quadratic control prob- lem consists of finding a control law minimizing a qua- dratic functional on trajectories of a linear dynamic object and has a solution if the total state vector of the object can be measured and if the model of the object is free of uncertainties (see, e.g., [1]). In [2, 3], the theory of linear matrix inequalities was used to solve this prob- lem in the case of an unmeasurable state when one can measure only the output vector of the object, which usually has a lower dimension than the state vector. In this paper, both of the above constraints are dropped. Specifically, the state of the object cannot be measured and the precise parameters of the object are unknown, while bounded sets of their admissible values are given instead. Under these assumptions, we give an upper bound for the objective functional for all admissible uncertainties and construct an optimal robust output controller that minimizes this bound. Let a controllable object be governed by the equa- tions (1) where x ∈ is the state; u ∈ is the control; y ∈ is the measurable output; Ω is the matrix corre- sponding to the unknown parameters and satisfying the condition (2) x˙ A F ΩE + ( ) x Bu , x 0 () + x 0 , = = y C 2 x , = R n x R n u R n y Ω T Ω η 2 I ; ≤ A, B, C 2 , F, and E are given matrices of suitable sizes; I is the identity matrix; and η > 0 is a given number. Moreover, suppose that we are given the functional (3) where z ∈ is the controllable output. Define a γ-optimal robust output control law of the form (4) such that, for all admissible Ω, closed-loop system (1), (4) is asymptotically stable and the target condition (5) is satisfied. Condition (5) can be interpreted as the damping of disturbances caused by deviations from the object’s ini- tial state down to the given level γ when the influence of a disturbance is estimated by the maximum ratio of the functional value to the squared norm of the initial state vector. Theorem. Given γ and some τ > 0, suppose that the matrices X 11 = > 0 and Y 11 = > 0 of order n x satisfy the linear matrix inequalities Ju () z 2 t , z d 0 ∞ ∫ C 1 x Du , + = = R n z x˙ r A r x r B r y , x r 0 () + 0, = = u C r x r D r y + = Ju () γ 2 x 0 2 x 0 ∀ 0 ≠ < X 11 T Y 11 T W 1 T A T X 11 X 11 A + C 1 T X 11 F η E T C 1 γτ I – 0 0 F T X 11 0 I – 0 η E 0 0 I – ⎝ ⎠ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ W 1 0, < CONTROL THEORY Optimal Robust Output Control D. V. Balandin a and M. M. Kogan b Presented by Academician S.K. Korovin April 11, 2007 Received April 12, 2007 DOI: 10.1134/S1064562407050389 a Department of Numerical and Functional Analysis, Nizhni Novgorod State University, ul. Ul’yanova 10, Nizhni Novgorod, 603005 Russia e-mail: balandin@pmk.unn.runnet.ru b Nizhni Novgorod State University of Architecture and Building, Il’inskaya ul. 65, Nizhni Novgorod, 603950 Russia e-mail: mkogan@nngasu.ru