Geodesic
On a singularly perturbed time-optimal control problem with two small parameters
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 2, pp. 76-92 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper we investigate a time-optimal control problem for a singularly perturbed linear autonomous system with two independent small parameters and smooth geometric constraints on the control in the form of a ball. The main difference of this case from the systems with fast and slow variables studied earlier is that here the matrix at the fast variables is a multidimensional analog of the second-order Jordan cell with zero eigenvalue and, thus, does not satisfy the standard condition of asymptotic stability. Continuing the research, we consider initial conditions depending on the second small parameter; in the degenerate case, this resulted in an asymptotic expansion of the solution of a fundamentally different type. The solvability of the problem is proved. We also derive and justify a complete power asymptotic expansion in the sense of Erdelyi of the optimal time and optimal control with respect to a small parameter at the derivatives in the equations of the systems.
Keywords: optimal control, time-optimal control problem, asymptotic expansion, singularly perturbed problem, small parameter. 
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A. R. Danilin; O. O. Kovrizhnykh. On a singularly perturbed time-optimal control problem with two small parameters. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 2, pp. 76-92. http://geodesic.mathdoc.fr/item/TIMM_2018_24_2_a8/

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