Control theory and dynamical systems are closely intertwined fields. Pontryagin's maximum principal offers a strong connection by providing a constructive way to synthesize feedback control laws via lifting the control system to a (Hamiltonian) dynamical system. Feedback laws can then be encoded as invariant manifolds of this induced dynamical system - under the condition that these manifolds project diffeomorphically back to the base control system. While feedback stabilization is impossible for many control systems, the above procedure can still be carried out. In this setting, the invariant manifold no longer projects diffeomorphically which results in the emergence of caustics. This paper offers an overview of the above connection by translating target sets from controls to isotropic submanifolds in symplectic geometry and associates feedback controllability to singularities of the induced invariant manifolds. Multiple low-dimensional examples are included to elucidate the theory.
We present a framework for the modeling and analysis of geometric hybrid dynamical systems as hybrid inclusions on $C^1$-manifolds. Using tools from nonsmooth and set-valued analysis on manifolds, we derive coordinate-independent sufficient conditions for the existence of nontrivial solutions to this type of systems. W...
Piyush P. Jirwankar, Daniel E. Ochoa, R. Sanfelice· 0 citations
Contraction analysis of dynamical systems turns a global convergence problem into a local one. If the variational system is stable in a uniform sense, then all trajectories of the original system converge toward each other. This paper develops a stochastic analogue for continuous-time stochastic systems, where converge...
We consider discrete-time dynamical systems generated by the iteration of asymptotically autonomous maps, whose asymptotic behavior may be either conservative or dissipative. These systems are of particular interest in the modeling of complex phenomena with time-dependent parameter variation. With an appropriate time c...
We develop an algebraic-dynamical correspondence for type-I oscillatory blow-up solutions of autonomous ODEs with asymptotically quasi-homogeneous vector fields. Periodic solutions of the balance law arising from asymptotic expansions of blow-ups are related to periodic orbits on the horizon for the desingularized vect...
We explore the problem of asymptotically stabilizing a class of interacting particle systems to a prescribed particle configuration with zero velocity. The class of interacting particle systems is motivated by the Cucker-Smale model on the mean-field level. Rather than designing a feedback by manually solving an IDA-PB...
This paper studies path-following control for a kinematic unicycle evolving on a smooth manifold embedded in \(\mathbb{R}^3\). The objective is to stabilize a one-dimensional path on the manifold without imposing a temporal parameterization, thereby distinguishing the problem from trajectory tracking. Working in a geom...
Adeel Akhtar, Mohamed Al Lawati· 1 citation
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