The problem of adaptive optimal control of a dynamical system, which belongs to the class of conditional variational problems with moving boundaries, is considered. A variational and computer study of the controlled adaptive motion of a material point is carried out for the problem of the energy quality functional minimizing with a moving, not predetermined right transboundary and in the case when the mass of the point changes depending on the unfixed final time. The problem is solved using the schemes and procedures of the classical calculus of variations, as well as adaptive estimation techniques, including the derivation of the variation of the auxiliary quality functional, the corresponding Euler equations, and the adaptive estimation algorithm. When solving a general conditional variational problem, the obtained closed system of differential equations was studied for the formation of an adaptive optimal control system for a dynamic plant with a given performance functional. The results of the unconditional formulation of the problem are generalized to the case of additional differential (nonholonomic) and holonomic constraints. In a variational adaptive optimal control problem, the transversality condition is formulated in terms of the local programming condition. The developed variational scheme of adaptive optimal synthesis can be used in the calculation and design of controlled dynamic systems. This optimization scheme is also promising for use in systems where operating time is non-fixed in advance. The results achieved in this paper concern obtaining specific equations, expressions, and formulas relative to the model example under study and finding graphs of the main time functions that determine the nature of the movement of the control object and the quality of the corresponding transients. The proposed adaptive optimal control algorithms for purposeful movement of the studied material point were tested in digital mode and showed their effectiveness which makes them promising for further use in more complex nonlinear adaptive systems of dynamic optimal control.