Measure-Driven Nonlinear Dynamic Systems with Applications to Optimal Impulsive Controls
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Measure-Driven Nonlinear Dynamic Systems with Applications to Optimal Impulsive Controls Nasir Uddin Ahmed1 · Shian Wang1,2 Received: 3 April 2019 / Accepted: 9 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we consider a class of nonlinear systems driven by measures generalizing the class of impulsive systems. We use measures as control and prove existence of optimal controls (measures) and present necessary conditions of optimality. We apply the general results to derive necessary conditions of optimality for purely impulsedriven systems. These results are then applied to optimal control problems related to geosynchronous satellites with some numerical results. Keywords Impulsive systems · Measures as controls · Existence of solutions · Existence of optimal controls · Necessary conditions of optimality · Practical applications Mathematics Subject Classification 34H05 · 49J15 · 49K15 · 93C15 · 93C95
1 Introduction An impulsive system is popularly described by a set of evolution equations on mutually disjoint intervals of time describing, on each interval, continuous evolution of state followed by a jump. To achieve a desired objective, one can use optimal control theory developed for impulsive systems to determine the jump sizes and jump instants. One of the effective computational methods is the control parameterization
Communicated by Emmanuel Trelat.
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Nasir Uddin Ahmed [email protected] Shian Wang [email protected]
1
University of Ottawa, Ottawa, ON, Canada
2
Present Address: University of Minnesota, Minneapolis, MN, USA
123
Journal of Optimization Theory and Applications
technique [1], which indirectly solves the problem using time scaling transformation and approximations [2]. In this paper, we consider a large class of nonlinear dynamic systems driven by measures generalizing the class of impulsive systems in the literature. We describe in one equation (instead of a set of equations) the general system subject to and controlled by measures generalizing impulsive forces. Moreover, we develop an algorithm to directly address the associated optimal control problem. Once a set of time points is specified, the algorithm can automatically select jump points to apply impulsive forces with appropriate sizes or eliminate unnecessary ones. This appears to be complementary to the approach proposed in [2]. However, it reduces the complexity without requiring time scaling transformation and constraint transcription. The remainder of the paper is organized as follows: In Sect. 2, we present the basic assumptions as well as the system model driven by signed measures and prove existence of solutions and regularity properties thereof. In Sect. 3, we present the optimal control problem. We prove the continuous dependence of solutions with respect to control measures and the existence of optimal controls. In Sect. 4, we present the necessary conditions of optimality. In Sect. 5, we consider the same dynamic system driven by positive measures with the cost
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