Geodesic
Sufficient Optimality Conditions for Hybrid Systems of Variable Dimension
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 308 (2020), pp. 88-100

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We consider an optimal control problem for a hybrid system whose continuous motion alternates with discrete variations (switchings) under which the dimension of the state space changes. The moments and the number of switchings are not specified in advance. They are determined as a result of minimizing a functional that incorporates the cost of each switching. The state space may change, for example, when the number of control objects varies, which is typical, in particular, of control problems for groups of a variable number of aircraft. We obtain sufficient optimality conditions for such systems and derive equations for the synthesis of optimal trajectories. The application of optimality conditions is demonstrated in academic examples. 
@article{TM_2020_308_a6,
     author = {A. S. Bortakovskii},
     title = {Sufficient {Optimality} {Conditions} for {Hybrid} {Systems} of {Variable} {Dimension}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {88--100},
     publisher = {mathdoc},
     volume = {308},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2020_308_a6/}
}
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A. S. Bortakovskii. Sufficient Optimality Conditions for Hybrid Systems of Variable Dimension. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 308 (2020), pp. 88-100. http://geodesic.mathdoc.fr/item/TM_2020_308_a6/