On Linearizability Conditions for Non-autonomous Control Systems

Solving the time-optimal control problem for nonlinear non-autonomous linearizable systems

“…In [5], linearizability conditions for nonlinear non-autonomous control systems were given; further analysis can be found in [10], [11]. In this subsection we formulate a direct corollary of these results related to a local statement of the problem.…”

“…Conditions 3 and 4 are specific for nonautonomous case [5], [11]. Condition 𝑎(𝑡, 𝑥) ∈ 𝐶 2 ((−𝛿, 𝛿) × 𝑈 (0)) is of technical character, see [10] for a detailed discussion.…”

“…Then, in Subsection 2.2, we recall some recent results on linearizability conditions for non-autonomous systems proposed in [5]. Additionally to linearizability conditions known since [6], [7] and generalized to systems of the class 𝐶 1 in [8], [9], in the non-autonomous case some new conditions arise, see [10] and [11] for further discussion.…”

Invariants of linear control systems with analytic matrices and the linearizability problem

“…This requirement is justified by the theorem on linearizability conditions, originally proposed and proved in [19]. Here we formulate the modification obtained in [21].…”

“…In this definition, the word "locally" recalls that the change of variables is only locally invertible in the general case and the word "analytically" means that a target nonautonomous linear system has analytic matrices. Definition 2 ([19,21]). We say that a system of the form (4) is locally analytically mappable in the domain Q on the time interval [α, β] to a preassigned linear controllable system (2) where A(t), b(t) are analytic on [α, β] if there exists a change of variables satisfying (9), ( 10) such that the system (4) in the new variables takes the form (2).…”