Informatics and Applications

2026, Volume 20, Issue 1, pp 2-11

MULTIPLICATIVE OUTPUT CONTROL UNDER A QUADRATIC CRITERION: DYNAMIC PROGRAMMING AND THE OPTIMAL SOLUTION

  • A. V. Bosov
  • I. V. Uryupin

Abstract

The paper addresses an optimal control problem for a quasi-linear output of a stochastic differential system driven by an Ito diffusion process. In contrast to the traditional additive control formulation, the controlled linear output is assumed to include multiplicative control resulting in a quasi-linear differential system with feedback. The problem is formulated using a general quadratic performance criterion which defines control objectives identical to those in the additive control model. This allows for a direct comparison of control strategies as alternative architectural solutions within the same application context. The study focuses on two multiplicative control configurations: one where the control acts as amultiplier of the system state, and another where itmultiplies an uncontrolled disturbance. The third possible case — output-multiplicative control — leads to a bilinear system; since its analysis requires a different methodological framework, it is excluded from this study. The solution is derived using a dynamic programming approach. Similar to the additive control case, the Bellman function is shown to take a quadratic form with respect to the output variable. However, in the state-multiplicative control configuration, the solution — characterized by three coefficients of the Bellman function — is substantially more complex. This complexity motivates the problem of synthesizing practically implementable approximations.While the disturbance-multiplicative control case is considerably simpler, its practical relevance is found to be limited.

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