Abstract
Geometrically, the affine connection is the main ingredient that underlies the covariant derivative, the parallel transport, the auto-parallel curves, the torsion tensor field, and the curvature tensor field on a finite-dimensional differentiable manifold. In this paper, we come up with a new idea of controllability and observability of states by using auto-parallel curves, and the minimum time problem controlled by the affine connection. The main contributions refer to the following: (i) auto-parallel curves controlled by a connection, (ii) reachability and controllability on the tangent bundle of a manifold, (iii) examples of equiaffine connections, (iv) minimum time problem controlled by a connection, (v) connectivity by stochastic perturbations of auto-parallel curves, and (vi) computing the optimal time and the optimal striking time. The connections with bounded pull-backs result in bang–bang optimal controls. Some significative examples on bi-dimensional manifolds clarify the intention of our paper and suggest possible applications. At the end, an example of minimum striking time with simulation results is presented.
Highlights
Geometers [1,2] used the affine connection only in problems of differential geometry, especially in order to introduce and analyze the covariant derivative, the parallel transport, the auto-parallel curves, the torsion tensor field, and the curvature tensor field on finitedimensional differentiable manifolds
The main contributions refer to the following: (i) autoparallel curves controlled by a connection, (ii) reachability and controllability on the tangent bundle of a manifold, (iii) examples of equiaffine connections, (iv) minimum time problem controlled by a connection, (v) connectivity by stochastic perturbations of auto-parallel curves, and (vi) computing the optimal time and the optimal striking time
We develop new ideas: (i) controllability and observability of states through auto-parallel curves, (ii) solving the minimum time problem controlled by the affine connection, and (iii) establishing stochastic connectivity via stochastic perturbations of auto-parallel Pfaff ODEs
Summary
Geometers [1,2] used the affine connection only in problems of differential geometry, especially in order to introduce and analyze the covariant derivative, the parallel transport, the auto-parallel curves, the torsion tensor field, and the curvature tensor field on finitedimensional differentiable manifolds. We emphasize that a connection controls the optimal time in which a given state is reached through an auto-parallel curve starting from a fixed point and by following the technique of Evans in the Lecture Notes [3]. The following issues are emphasized: Pfaff form of auto-parallel ODEs, stochastic perturbations, the simplified stochastic minimum principle, and striking time. Bang–bang type controls arise in many applications, for example, applied physics [4,5], game theory [6], chemistry [7], biology [8], socio-economic systems [9], minimum-time problems [10], etc. We recall that there exist constant connections that are solutions for curvature-flatness PDEs [15,16]
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