On the potentiality of a class of operators relative to local bilinear forms. (English) Zbl 1477.49054

Summary: The inverse problem of the calculus of variations (IPCV) is solved for a second-order ordinary differential equation with the use of a local bilinear form. We apply methods of analytical dynamics, nonlinear functional analysis, and modern methods for solving the IPCV. In the paper, we obtain necessary and sufficient conditions for a given operator to be potential relative to a local bilinear form, construct the corresponding functional, i.e., found a solution to the IPCV, and define the structure of the considered equation with the potential operator. As a consequence, similar results are obtained when using a nonlocal bilinear form. Theoretical results are illustrated with some examples.


49N45 Inverse problems in optimal control
49J05 Existence theories for free problems in one independent variable
49J15 Existence theories for optimal control problems involving ordinary differential equations
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