Two classes of optimal observation problems arising in the context of optimal robust control of linear systems under inaccurate and incomplete state measurements are investigated. Linear optimal observation problems are associated with the construction of optimal guaranteed open-loop controls and provide estimates of information sets that describe states consistent with measurements obtained. A priori optimal observation problems are related to the synthesis of multiple-closable control strategies and the determination of the worst-case future information sets. These problems are formulated as specific multi-level mathematical programming problems, for which a recurrent procedure for their reduction to large-scale linear programming problems is developed. The proposed algorithms are oriented towards real-time applications in optimal guaranteed control systems.

1. Kurzhanski A. B. Control and Observation under Uncertainty. Moscow, Nauka Publ., 1977. 392 p. (in Russian).

2. Chernousko F. L. State Estimation for Dynamic Systems. The Ellipsoid Method. Moscow, Nauka Publ., 1988. 320 p. (in Russian).

3. Kostousova E. K. On polyhedral estimation of reachability sets of linear multistep systems. Automation and Remote Control, 1997, vol. 58, iss. 3, pp. 374–382.

4. Kostousova E. K. State estimation for dynamic systems via parallelotopes: optimization and parallel computations. Optimization Methods & Software, 1998, vol. 9, no. 14, pp. 269–306 (in Russian).

5. Gabasov R., Kirillova F. M. Dual problems of control, observation and identification. Doklady Akademii Nauk BSSR, 1990, vol. 34, no. 9, pp. 777–780 (in Russian).

6. Gabasov R., Dmitruk N. M., Kirillova F. M. Optimal control of multidimensional systems from inaccurate measurements of their output signals. Proceedings of the Steklov Institute of Mathematics, 2004, suppl. 2, pp. S52–S75.

7. Dmitruk N. M. Optimal robust control of linear systems from inaccurate measurements of output signals. Journal of Automation and Information Sciences, 2011, vol. 43, iss. 11, pp. 30–47.

8. Dmitruk N. M. Optimal closed-loop measurement feedback in a linear optimal control problem. Proceedings of the Steklov Institute of Mathematics, 2025, vol. 329, pp. S77–S92.

9. Gabasov R., Dmitruk N. M., Kirillova F. M. Optimal observation of nonstationary systems. Journal of Computer and Systems Sciences International, 2002, vol. 41, no. 2, pp. 195–206.

10. Gabasov R., Dmitruk N. M., Kirillova F. M. Optimal conditional relative observation of non-stationary linear systems. Journal of Applied Mathematics and Mechanics, 2004, vol. 68, no. 4, pp. 523–536.

11. Kurzhanski A. B., Varaiya P. On reachability under uncertainty. SIAM Journal on Control and Optimization, 2002, vol. 41, no. 1, pp. 181–216.

12. Dmitruk N. M. Multiply closed control strategy in a linear terminal problem of optimal guaranteed control. Proceedings of the Steklov Institute of Mathematics, 2022, vol. 319, suppl. 1, pp. S112– S128.

13. Gabasov R., Kirillova F. M. Constructive Methods of Optimization. Part 2. Control Problems. Minsk, Universitetskoe, 1984. 207 p. (in Russian).

14. Gabasov R., Dmitruk N. M., Kirillova F. M. Optimal control under persistent disturbances. Proceedings of the Steklov Institute of Mathematics, 2010, vol. 269, suppl. 1, pp. S52–S75.

15. Gabasov R., Kirillova F. M. Optimization of Linear Systems. Minsk, BSU, 1973. 246 p. (in Russian).

16. Boyd S. P., Vandenberghe L. Convex Optimization. Cambridge, Cambridge University Press, 2023. 716 p.