Showing papers in "Journal of Dynamical and Control Systems in 2022"

A Gradient Flow Equation for Optimal Control Problems With End-point Cost

Abstract: Abstract In this paper, we consider a control system of the form $\dot x = F(x)u$ x ̇ = F ( x ) u , linear in the control variable u . Given a fixed starting point, we study a finite-horizon optimal control problem, where we want to minimize a weighted sum of an end-point cost and the squared 2-norm of the control. This functional induces a gradient flow on the Hilbert space of admissible controls, and we prove a convergence result by means of the Lojasiewicz-Simon inequality. Finally, we show that, if we let the weight of the end-point cost tend to infinity, the resulting family of functionals is Γ -convergent, and it turns out that the limiting problem consists in joining the starting point and a minimizer of the end-point cost with a horizontal length-minimizer path.

Lipschitz Carnot-Carathéodory Structures and their Limits

Abstract: Abstract In this paper we discuss the convergence of distances associated to converging structures of Lipschitz vector fields and continuously varying norms on a smooth manifold. We prove that, under a mild controllability assumption on the limit vector-fields structure, the distances associated to equi-Lipschitz vector-fields structures that converge uniformly on compact subsets, and to norms that converge uniformly on compact subsets, converge locally uniformly to the limit Carnot-Carathéodory distance. In the case in which the limit distance is boundedly compact, we show that the convergence of the distances is uniform on compact sets. We show an example in which the limit distance is not boundedly compact and the convergence is not uniform on compact sets. We discuss several examples in which our convergence result can be applied. Among them, we prove a subFinsler Mitchell’s Theorem with continuously varying norms, and a general convergence result for Carnot-Carathéodory distances associated to subspaces and norms on the Lie algebra of a connected Lie group.

Exact Boundary Controllability of the Linear Biharmonic Schrödinger Equation with Variable Coefficients

Abstract: In this paper, we study the exact boundary controllability of the linear fourth-order Schrödinger equation, with variable physical parameters and clamped boundary conditions on a bounded interval. The control acts on the first spatial derivative at the right endpoint. We prove that this control system is exactly controllable at any time $$T>0$$ . The proofs are based on a detailed spectral analysis and the use of nonharmonic Fourier series.

A Geometric Criterion for the Existence of Chaos Based on Periodic Orbits in Continuous-Time Autonomous Systems

Abstract: A new geometric criterion is derived for the existence of chaos in continuous-time autonomous systems in three-dimensional Euclidean spaces, where a type of Smale horseshoe in a subshift of finite type exists, but the intersection of stable and unstable manifolds of two points on a hyperbolic periodic orbit does not imply the existence of a Smale horseshoe of the same type on any cross section of these two points. This criterion is based on the existence of a hyperbolic periodic orbit, differing from the classical equilibrium-based Shilnikov criterion and the condition of transversal homoclinic or heteroclinic orbit of a Poincaré map.

Quenched Decay of Correlations for One-dimensional Random Lorenz Maps

Abstract: Abstract We study rates of mixing for small random perturbations of one-dimensional Lorenz maps. Using a random tower construction, we prove that, for Hölder observables, the random system admits exponential rates of quenched correlation decay.

On the Structure of Singular Points of a Solution to Newton’s Least Resistance Problem

Abstract: We consider the following problem stated in 1993 by Buttazzo and Kawohl (Math Intell 15:7–12, 1993): minimize the functional $${\int \limits } {\int \limits }_{\Omega } (1 + | abla u(x,y)|^{2})^{-1} dx dy$$ in the class of concave functions u : Ω → [0,M], where $${\Omega } \subset \mathbb {R}^{2}$$ is a convex domain and M > 0. It generalizes the classical minimization problem, which was initially stated by I. Newton in 1687 in the more restricted class of radial functions. The problem is not solved until now; there is even nothing known about the structure of singular points of a solution. In this paper we, first, solve a family of auxiliary 2D least resistance problems and, second, apply the obtained results to study singular points of a solution to our original problem. More precisely, we derive a necessary condition for a point being a ridge singular point of a solution and prove, in particular, that all ridge singular points with horizontal edge lie on the top level and zero level sets.

Differentiability of the Diffusion Coefficient for a Family of Intermittent Maps

Abstract: Abstract It is well known that the Liverani–Saussol–Vaienti map satisfies a central limit theorem for Hölder observables in the parameter regime where the correlations are summable. We show that when C 2 observables are considered, the variance of the limiting normal distribution is a C 1 function of the parameter. We first show this for the first return map to the base of the second branch by studying the Green-Kubo formula, then conclude the result for the original map using Kac’s lemma and relying on linear response.

Optimization of Adiabatic Control Strategies Along Non-mixing Curves with Singularities

Abstract: In this paper, we consider a system driven by a controlled Schrödinger equation with two external control inputs. Motivated by applications to the control of quantum systems having conical or semi-conical eigenvalue intersections, we propose to study the singularities and the parametric bifurcations of the associated non-mixing field, along whose integral curves in the space of controls the adiabatic approximation holds with higher precision. Our results can be applied to optimize the adiabatic control strategies of well known quantum systems such as Qubit systems, Stirap Processes and Eberly-Law models.

Operators Arising as Second Variation of Optimal Control Problems and Their Spectral Asymptotics

Abstract: Abstract We compute the asymptotic for the eigenvalues of a particular class of compact operators deeply linked with the second variation of optimal control problems. We characterize this family in terms of a set of finite dimensional data and we apply this results to a particular class of singular extremal to get a nice description of the spectrum of the second variation.

A relation between entropy and transitivity of Anosov diffeomorphisms

Abstract: It is known that transitive Anosov diffeomorphisms have a unique measure of maximal entropy (MME). Here we discuss the converse question. Transitivity of Anosov diffeomorphisms can be reached under suitable hypotheses on Lyapunov exponents on the set of periodic points and the structure of the MME. In another way, assuming together the uniqueness of MME and that every point is regular, in Oseledec’s Theorem sense, also we can get the transitivity of Anosov diffeomorphisms in this setting.

Some Variational Principles for the Metric Mean Dimension of a Semigroup Action

Abstract: In this manuscript, we show that the metric mean dimension of a semigroup action satisfies three variational principles: (a) in our first result, we consider the local entropy function for a free semigroup action and show that the metric mean dimension satisfies a variational principle in terms of such function; (b) the second one is about a definition of Katok’s entropy for a free semigroup action introduced in Carvalho et al. (Ergod, 42, 65–85, 2022); (c) in our third result, based on the definition of Shapira’s entropy, introduced in Shapira (Israel J Math, 158, 225–247, 2007) for a single dynamic, we extend the definition of Shapira’s entropy for a semigroup action. We also obtain a formula which relates the Shapira’s entropy of a free semigroup action and the Shapira’s entropy of the induced skew product; (d) in our fourth result, we obtain a variational principle involving the metric mean dimension and the Shapira’s entropy of a free semigroup action; (e) in the last two theorems, we extend the definition of metric mean dimension and the topological entropy when we have a finitely generated semigroup inspired in the definition of topological entropy introduced in Ghys et al. (Acta Math, 160, 105–142, 1988). In this context, we obtain a partial variational principle for the metric mean dimension. Our results are inspired in the ones obtained by (Lindenstrauss et al. 2019) and Velozo and Velozo (2017) and Gutman and Sṕiewak (Stud Math, 261, 345–360, 2021) and Shi (IEEE Trans Inf Theory, 68, 4282–4288, 2022).

The General Solution for Affine Control Systems on Lie Groups

Abstract: The purpose of this paper is to present explicitly the solution curve for commutative affine control systems on Lie groups under the assumption that the automorphisms associated with the linear vector fields commutes. If we assume that the derivations associated with the linear vector fields of the system are inner, we obtain a simpler solution and we show some results of controllability. To finish, we work with conjugation by homomorphism of Lie groups between affine systems.