Geodesic
Optimal adiabatic compression of sphere layers of gas
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 1 (1992), pp. 204-212

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Approximate laws of control of unshock compression of homogeneous sphere layers of polytropic gas, that require minimum rate of energy flow to achieve a given degree of compression, are analytically constructed. It is shown that the optimal law of control has only one point of switching and consists of two steps. At the first step a compressing piston moves at the maximum rate, and the law of rate alteration is found with the use of characteristic series. At the second step the law of control is found in explicit form. 
@article{TIMM_1992_1_a15,
     author = {A. F. Sidorov},
     title = {Optimal adiabatic compression of sphere layers of gas},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {204--212},
     publisher = {mathdoc},
     volume = {1},
     year = {1992},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_1992_1_a15/}
}
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A. F. Sidorov. Optimal adiabatic compression of sphere layers of gas. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 1 (1992), pp. 204-212. http://geodesic.mathdoc.fr/item/TIMM_1992_1_a15/