We study the set of T-periodic solutions of a class of T-periodically perturbed coupled and nonau... more We study the set of T-periodic solutions of a class of T-periodically perturbed coupled and nonautonomous differential equations on manifolds. By using degree-theoretic methods we obtain a global continuation result for the T-periodic solutions of the considered equations.
We apply topological methods to obtain global continuation results for harmonic solutions of some... more We apply topological methods to obtain global continuation results for harmonic solutions of some periodically perturbed ordinary differential equations on a k-dimensional differentiable manifold M ⊆ R m. We assume that M is globally defined as the zero set of a smooth map and, as a first step, we determine a formula which reduces the computation of the degree of a tangent vector field on M to the Brouwer degree of a suitable map in R m. As further applications, we study the set of harmonic solutions to periodic semi-explicit differential-algebraic equations.
This paper gives sufficient conditions for a class of bang-bang extremals with multiple switches ... more This paper gives sufficient conditions for a class of bang-bang extremals with multiple switches to be locally optimal in the strong topology. The conditions are the natural generalizations of the ones considered in [5, 14] and [16]. We require both the strict bang-bang Legendre condition, and the second order conditions for the finite dimensional problem obtained by moving the switching times of the reference trajectory.
Electronic Journal of Qualitative Theory of Differential Equations, 2023
Using a topological approach we investigate the structure of the set of forced oscillations of a ... more Using a topological approach we investigate the structure of the set of forced oscillations of a class of reducible second order functional retarded differential equations subject to periodic forcing. More precisely, we consider a delay-type functional dependence involving a gamma probability distribution and, using a linear chain trick, we formulate a first order system of ODEs whose T-periodic solutions correspond to those of the functional equation.
We consider the propagation of surface shear waves in a half-plane, whose shear modulus µ(y) and ... more We consider the propagation of surface shear waves in a half-plane, whose shear modulus µ(y) and density ρ(y) depend continuously on the depth coordinate y. The problem amounts to studying the parametric Sturm-Liouville equation on a half-line with frequency ω and wave number k as the parameters. The Neumann (traction-free) boundary condition and the requirement of decay at infinity are imposed. The condition of solvability of the boundary value problem determines the dispersion spectrum ω(k) for the corresponding surface wave. We establish the criteria for non-existence of surface waves and for the existence of N (k) surface wave solutions, with N (k) → ∞ as k → ∞. The most intriguing result is a possibility of the existence of infinite number of solutions, N (k) = ∞, for any given k. These three options are conditioned by the properties of µ(y) and ρ(y). functionally graded medium and surface shear waves and parametric Sturm-Liouville problem
Mathematical Methods in The Applied Sciences, Jan 9, 2018
We find multiplicity results for forced oscillations of a periodically perturbed autonomous secon... more We find multiplicity results for forced oscillations of a periodically perturbed autonomous second order equation, the perturbing term possibly depending on the whole history of the system. The techniques that we employ are topological in nature but the technical details are hidden in the proofs and completely transparent to the reader only interested in the results.
Communications in Contemporary Mathematics, Aug 24, 2021
We provide some new estimates for the topological degree of a class of continuous and piecewise l... more We provide some new estimates for the topological degree of a class of continuous and piecewise linear maps based on a classical integral computation formula. We provide applications to some nonlinear problems that exhibit a local [Formula: see text] structure.
We study, by means of a topological approach, the forced oscillations of second order functional ... more We study, by means of a topological approach, the forced oscillations of second order functional retarded differential equations subject to periodic perturbations. We consider a delay-type functional dependence involving a gamma probability distribution. By a linear chain trick we obtain a first order system of ODE's whose T-periodic solutions correspond to those of the functional equation.
We consider a general class of nonlinear dynamical systems with memory.We prove under which condi... more We consider a general class of nonlinear dynamical systems with memory.We prove under which conditions periodic perturbations allow solutions with the same period. In particular, we consider memory kernels of $\Gamma$-type, which give prominence to critical phenomena in the past.
Linear systems with controllable (A, B) and bounded control range have a unique control set. This... more Linear systems with controllable (A, B) and bounded control range have a unique control set. This control set is bounded if and only if A is hyperbolic. Then uniqueness also remains valid under small nonlinear perturbations. Examples show that for nonhyperbolic A small nonlinear perturbations may lead to infinitely many (invariant) control sets.
Journal of Dynamics and Differential Equations, 2021
We study the existence of a connected “branch” of periodic solutions of T-periodic perturbations ... more We study the existence of a connected “branch” of periodic solutions of T-periodic perturbations of a particular class of functional differential equations on differentiable manifolds. Our result is obtained by a combination of degree-theoretic methods and a technique that allows to associate the bounded solutions of the functional equation to bounded solutions of a suitable ordinary differential equation.
International Journal of Heat and Mass Transfer, 2021
Abstract We compare traveling-wave-type solutions to Maxwell-Cattaneo’s description of heat trans... more Abstract We compare traveling-wave-type solutions to Maxwell-Cattaneo’s description of heat transfer with those of Capriz-Wilmanski-Mariano’s version, which includes nonlinearities due to microstructural contributions in the presence of heat-flux-driven phase transition in a conductor otherwise rigid. Our closed-form results for the nonlinear scheme, based on asymptotic-type analysis, show how temperature traveling-wave-type evolution is perturbed by the influence of microstructural effects. Also, our results underline how even in Maxwell-Cattaneo’s scheme initial conditions in terms of heat flux and temperature have some hidden link, although in the scheme these variables are a priori independent with each other.
We study the set of T-periodic solutions of a class of T-periodically perturbed coupled and nonau... more We study the set of T-periodic solutions of a class of T-periodically perturbed coupled and nonautonomous differential equations on manifolds. By using degree-theoretic methods we obtain a global continuation result for the T-periodic solutions of the considered equations.
Lecture Notes in Control and Information Sciences, 1999
Linear systems with controllable (A, B) and bounded control range have a unique control set. This... more Linear systems with controllable (A, B) and bounded control range have a unique control set. This control set is bounded if and only if A is hyperbolic. Then uniqueness also remains valid under small nonlinear perturbations. Examples show that for nonhyperbolic A small nonlinear perturbations may lead to infinitely many (invariant) control sets.
The controllability behavior of nonlinear control systems is described by associating semigroups ... more The controllability behavior of nonlinear control systems is described by associating semigroups to locally maximal subsets of complete controllability, i.e., local control sets. Periodic trajectories are called equivalent if there is a 'homotopy' between them involving only trajectories. The resulting object is a semigroup, which we call the dynamic index of the local control set. It measures the different ways the system can go through the local control set.
In this paper we associate a semigroup to a locally maximal subset of complete controllability, i... more In this paper we associate a semigroup to a locally maximal subset of complete controllability, i.e., a local control set. This fundamental semigroup is based on equivalence classes under homotopies in the set of trajectories. It reflects the structure of the set of closed (trajectory) loops in the local control set. We discuss the relations between different local control sets and prove a Van Kampen-type theorem for their unions and intersections.
The controllability behavior of nonlinear control systems is described by associating semigroups ... more The controllability behavior of nonlinear control systems is described by associating semigroups to locally maximal subsets of complete controllability, i.e., local control sets. Periodic trajectories are called equivalent if there is a 'homotopy' between them involving only trajectories. The resulting object is a semigroup, which we call the dynamic index of the local control set. It measures the different ways the system can go through the local control set. A number of examples are considered.
We study the set of T-periodic solutions of a class of T-periodically perturbed coupled and nonau... more We study the set of T-periodic solutions of a class of T-periodically perturbed coupled and nonautonomous differential equations on manifolds. By using degree-theoretic methods we obtain a global continuation result for the T-periodic solutions of the considered equations.
We apply topological methods to obtain global continuation results for harmonic solutions of some... more We apply topological methods to obtain global continuation results for harmonic solutions of some periodically perturbed ordinary differential equations on a k-dimensional differentiable manifold M ⊆ R m. We assume that M is globally defined as the zero set of a smooth map and, as a first step, we determine a formula which reduces the computation of the degree of a tangent vector field on M to the Brouwer degree of a suitable map in R m. As further applications, we study the set of harmonic solutions to periodic semi-explicit differential-algebraic equations.
This paper gives sufficient conditions for a class of bang-bang extremals with multiple switches ... more This paper gives sufficient conditions for a class of bang-bang extremals with multiple switches to be locally optimal in the strong topology. The conditions are the natural generalizations of the ones considered in [5, 14] and [16]. We require both the strict bang-bang Legendre condition, and the second order conditions for the finite dimensional problem obtained by moving the switching times of the reference trajectory.
Electronic Journal of Qualitative Theory of Differential Equations, 2023
Using a topological approach we investigate the structure of the set of forced oscillations of a ... more Using a topological approach we investigate the structure of the set of forced oscillations of a class of reducible second order functional retarded differential equations subject to periodic forcing. More precisely, we consider a delay-type functional dependence involving a gamma probability distribution and, using a linear chain trick, we formulate a first order system of ODEs whose T-periodic solutions correspond to those of the functional equation.
We consider the propagation of surface shear waves in a half-plane, whose shear modulus µ(y) and ... more We consider the propagation of surface shear waves in a half-plane, whose shear modulus µ(y) and density ρ(y) depend continuously on the depth coordinate y. The problem amounts to studying the parametric Sturm-Liouville equation on a half-line with frequency ω and wave number k as the parameters. The Neumann (traction-free) boundary condition and the requirement of decay at infinity are imposed. The condition of solvability of the boundary value problem determines the dispersion spectrum ω(k) for the corresponding surface wave. We establish the criteria for non-existence of surface waves and for the existence of N (k) surface wave solutions, with N (k) → ∞ as k → ∞. The most intriguing result is a possibility of the existence of infinite number of solutions, N (k) = ∞, for any given k. These three options are conditioned by the properties of µ(y) and ρ(y). functionally graded medium and surface shear waves and parametric Sturm-Liouville problem
Mathematical Methods in The Applied Sciences, Jan 9, 2018
We find multiplicity results for forced oscillations of a periodically perturbed autonomous secon... more We find multiplicity results for forced oscillations of a periodically perturbed autonomous second order equation, the perturbing term possibly depending on the whole history of the system. The techniques that we employ are topological in nature but the technical details are hidden in the proofs and completely transparent to the reader only interested in the results.
Communications in Contemporary Mathematics, Aug 24, 2021
We provide some new estimates for the topological degree of a class of continuous and piecewise l... more We provide some new estimates for the topological degree of a class of continuous and piecewise linear maps based on a classical integral computation formula. We provide applications to some nonlinear problems that exhibit a local [Formula: see text] structure.
We study, by means of a topological approach, the forced oscillations of second order functional ... more We study, by means of a topological approach, the forced oscillations of second order functional retarded differential equations subject to periodic perturbations. We consider a delay-type functional dependence involving a gamma probability distribution. By a linear chain trick we obtain a first order system of ODE's whose T-periodic solutions correspond to those of the functional equation.
We consider a general class of nonlinear dynamical systems with memory.We prove under which condi... more We consider a general class of nonlinear dynamical systems with memory.We prove under which conditions periodic perturbations allow solutions with the same period. In particular, we consider memory kernels of $\Gamma$-type, which give prominence to critical phenomena in the past.
Linear systems with controllable (A, B) and bounded control range have a unique control set. This... more Linear systems with controllable (A, B) and bounded control range have a unique control set. This control set is bounded if and only if A is hyperbolic. Then uniqueness also remains valid under small nonlinear perturbations. Examples show that for nonhyperbolic A small nonlinear perturbations may lead to infinitely many (invariant) control sets.
Journal of Dynamics and Differential Equations, 2021
We study the existence of a connected “branch” of periodic solutions of T-periodic perturbations ... more We study the existence of a connected “branch” of periodic solutions of T-periodic perturbations of a particular class of functional differential equations on differentiable manifolds. Our result is obtained by a combination of degree-theoretic methods and a technique that allows to associate the bounded solutions of the functional equation to bounded solutions of a suitable ordinary differential equation.
International Journal of Heat and Mass Transfer, 2021
Abstract We compare traveling-wave-type solutions to Maxwell-Cattaneo’s description of heat trans... more Abstract We compare traveling-wave-type solutions to Maxwell-Cattaneo’s description of heat transfer with those of Capriz-Wilmanski-Mariano’s version, which includes nonlinearities due to microstructural contributions in the presence of heat-flux-driven phase transition in a conductor otherwise rigid. Our closed-form results for the nonlinear scheme, based on asymptotic-type analysis, show how temperature traveling-wave-type evolution is perturbed by the influence of microstructural effects. Also, our results underline how even in Maxwell-Cattaneo’s scheme initial conditions in terms of heat flux and temperature have some hidden link, although in the scheme these variables are a priori independent with each other.
We study the set of T-periodic solutions of a class of T-periodically perturbed coupled and nonau... more We study the set of T-periodic solutions of a class of T-periodically perturbed coupled and nonautonomous differential equations on manifolds. By using degree-theoretic methods we obtain a global continuation result for the T-periodic solutions of the considered equations.
Lecture Notes in Control and Information Sciences, 1999
Linear systems with controllable (A, B) and bounded control range have a unique control set. This... more Linear systems with controllable (A, B) and bounded control range have a unique control set. This control set is bounded if and only if A is hyperbolic. Then uniqueness also remains valid under small nonlinear perturbations. Examples show that for nonhyperbolic A small nonlinear perturbations may lead to infinitely many (invariant) control sets.
The controllability behavior of nonlinear control systems is described by associating semigroups ... more The controllability behavior of nonlinear control systems is described by associating semigroups to locally maximal subsets of complete controllability, i.e., local control sets. Periodic trajectories are called equivalent if there is a 'homotopy' between them involving only trajectories. The resulting object is a semigroup, which we call the dynamic index of the local control set. It measures the different ways the system can go through the local control set.
In this paper we associate a semigroup to a locally maximal subset of complete controllability, i... more In this paper we associate a semigroup to a locally maximal subset of complete controllability, i.e., a local control set. This fundamental semigroup is based on equivalence classes under homotopies in the set of trajectories. It reflects the structure of the set of closed (trajectory) loops in the local control set. We discuss the relations between different local control sets and prove a Van Kampen-type theorem for their unions and intersections.
The controllability behavior of nonlinear control systems is described by associating semigroups ... more The controllability behavior of nonlinear control systems is described by associating semigroups to locally maximal subsets of complete controllability, i.e., local control sets. Periodic trajectories are called equivalent if there is a 'homotopy' between them involving only trajectories. The resulting object is a semigroup, which we call the dynamic index of the local control set. It measures the different ways the system can go through the local control set. A number of examples are considered.
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Papers by Marco Spadini