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Maximal deviation problem, absolute stability and robust stabilization

In 1939 B.V.Bulgakov posed the problem of maximal deviation (added disturbances) for linear stationary systems for the case of one coordinate and a fixed terminal moment. This problem arose in connection with an urgent technical problem of ballistic deviation estimation of a gyrocompass on a maneuvering ship. The problem was the first extreme problem on a functional set with restrictions of inequality type and in this preceded all other problems of optimum control with restrictions. A solution obtained by B.V.Bulgakov shows that the Pontryagin maximum principle in this case is not only necessary, but also a sufficient condition of optimality. Further development of this problem was done by J.N.Rojtenberg's, L.S. Gnoenskiy, V.V.Aleksandrov, L.N.Tarakanova, etc.

The problem of maximal deviations was modified for the case of non-fixed time and parametrical disturbances (V.V.Alexandrov). This enabled to develop a new variational approach to absolute stability. Formulation of the absolute stability problem is also connected with B.V.Bulgakov. In 1942 it for the first time he gave a description of uncertainties arising in stabilizing servo systems as a functional set. The variational method in absolute stability allows to reduce the problem for oscillatory systems to a problem of maximal deviation on a half-cycle of the corresponding quadratic form which in its essence is the fluctuating Lyapunov's function (V.V.Alexandrov, V.I.Zhermolenko, etc.).

Further development of this method is connected with the minimax technique of robust stabilization (V.V.Alexandrov, V.I.Zhermolenko) and maximin testing of accuracy of robust stabilization (V.V.Alexandrov). The maximin testing problem together with the minimax problem of robust stabilization forms a dynamic game, and allows to find an objective estimate of stabilization quality of non-stationary motions. When a saddle point exists, the minimax strategy of robust stabilization is found on the way. This technique was also applied to stochastic systems. Note that the procedure of testing does not demand knowledge of the stabilization algorithm, and thus is suitable not only to qualify an algorithm in an onboard computer, but in the central nervous system of the operator as well (for example, in the case of remote manual control) (V.A.Sadovnichy, V.V.Alexandrov, S.S.Lemak).

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