This paper addresses the tempered dichotomy for dynamical systems, generated by the random linear infinite-dimensional equations over measure-preserving dynamical systems (Ω , ℱ, P, θ ) on the half-line [0,∞ ) . We prove that the solutions of u̇=A(θ _t)u+f, t≥ 0 belong to ℒ^∞ _1 for any f∈ℒ^p_K if and only if the homogenous system u̇=A(θ _t)u has a tempered exponential dichotomy on the half-line ℝ_+ .
The main purpose of this paper is to provide an explicit description of the invariant control sets for a class of control systems induced on the unit quaternion sphere $$S^3$$ S 3 by the action of the Lorentz group $$\textrm{SO}(1,4)$$ SO ( 1 , 4 ) and then generalize it to the sphere $$S^{n-1}$$ S n - 1 . These control sets are the maximal subsets of approximate controllability for the control systems. Describing them in detail is generally challenging due to the complexity of the geometry and topology of the underlying differentiable manifold and the behavior of the vector fields defining the control system. In this work, the Lie theory and the quaternions play a fundamental role in achieving our main results.
We compensate for the scalability issues in controller synthesis by developing a hierarchical control scheme within the framework of ( $$\upgamma ,\updelta $$ γ , δ )-similarity, which measures to what extent a potentially non-deterministic system satisfies specifications expressed as solution trajectories of a dynamical ‘specification’ system. This scheme synthesizes a controller for a non-deterministic ‘concrete’ system in three hierarchical steps. First, an ‘abstract’ system, which is a low-dimensional model of the concrete system, is obtained. Then, a controller is designed for the abstract system. At last, the abstract controller is refined into the concrete controller through an ‘interface’. To enable this, we introduce and characterize the notion of ( $$\upgamma ,\updelta $$ γ , δ )-abstraction that utilizes an $$\mathcal {L}_2$$ L 2 approximation metric to measure the behavioral similarity of the concrete system and its abstraction in the presence of the interface. We utilize this characterization to propose a step-by-step procedure to construct the interface. We then synthesize the abstract controller and refine it into a concrete one.
