R.V. Gamkrelidze's maximum principle for optimal control problems with bounded phase coordinates and its relation to other optimality conditions

Necessary optimality conditions in optimal control problems with state constraints in the form of Pontryagin's maximum principle (MP) are studied. all the functions involved in the formulation of the problem are continuously differentiable, while vector function is twice continuously differentiable. An admissible process is said to be regular if there exists a number and bounded functions. Each of the functions is constant on any time interval where the optimal trajectory lies entirely in the interior of the set defined by the jth state constraint. For an optimal process in the problem, it is assumed that the terminal constraints at the point are regular, the phase and mixed constraints are regular, and the state constraints are compatible with the terminal ones at the point. For an admissible process satisfying the MP, it is assumed that the terminal constraints at the point are regular, the mixed constraints are regular, the state constraints are compatible with the terminal ones.