Cauchy Problems for Hamilton—Jacobi Equations
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In the framework of the proposed approach, the existence and uniqueness of minimax solution for a wide class of boundary-value problems and Cauchy problems can be proved. The Cauchy problem for Hamilton-Jacobi equation is examined in this chapter. Proofs of uniqueness and existence theorems are based on the property of weak invariance of minimax solutions with respect to characteristic inclusions. These inclusions are considered in the present section. We formulate also equivalent definitions of minimax solutions of Hamilton-Jacobi equations. Uniqueness and existence theorems are proved in the next sections. It can be seen from the proofs that these theorems actually provide criteria for the stability of solutions with respect to small perturbations of the Hamiltonian and the terminal function.
KeywordsCauchy Problem Viscosity Solution Lower Semicontinuous Differential Game Differential Inclusion
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