On linear-quadratic dissipative control processes with time-varying coefficients
Roberta Fabbri Russell Johnson Sylvia Novo Carmen Núñez
Discrete & Continuous Dynamical Systems - A 2013, 33(1): 193-210 doi: 10.3934/dcds.2013.33.193
Yakubovich, Fradkov, Hill and Proskurnikov have used the Yaku-bovich Frequency Theorem to prove that a strictly dissipative linear-quadratic control process with periodic coefficients admits a storage function, and various related results. We extend their analysis to the case when the coefficients are bounded uniformly continuous functions.
keywords: dissipativity Linear-quadratic control system supply rate storage function.
Null controllable sets and reachable sets for nonautonomous linear control systems
Roberta Fabbri Sylvia Novo Carmen Núñez Rafael Obaya
Discrete & Continuous Dynamical Systems - S 2016, 9(4): 1069-1094 doi: 10.3934/dcdss.2016042
Under the assumption of lack of uniform controllability for a family of time-dependent linear control systems, we study the dimension, topological structure and other dynamical properties of the sets of null controllable points and of the sets of reachable points. In particular, when the space of null controllable vectors has constant dimension for all the systems of the family, we find a closed invariant subbundle where the uniform null controllability holds. Finally, we associate a family of linear Hamiltonian systems to the control family and assume that it has an exponential dichotomy in order to relate the space of null controllable vectors to one of the Lagrange planes of the continuous hyperbolic splitting.
keywords: reachable sets abnormal systems proper focal points. null controllable sets rotation number linear Hamiltonian systems Nonautonomous linear control systems
Russell Johnson Roberta Fabbri Sylvia Novo Carmen Núñez Rafael Obaya
Discrete & Continuous Dynamical Systems - S 2016, 9(4): i-iii doi: 10.3934/dcdss.201604i
Generally speaking, the term nonautonomous dynamics refers to the systematic use of dynamical tools to study the solutions of differential or difference equations with time-varying coefficients. The nature of the time variance may range from periodicity at one extreme, through Bohr almost periodicity, Birkhoff recurrence, Poisson recurrence etc. to stochasticity at the other extreme. The ``dynamical tools'' include almost everywhere Lyapunov exponents, exponential splittings, rotation numbers, and the theory of cocycles, but are by no means limited to these. Of course in practise one uses whatever ``works'' in the context of a given problem, so one usually finds dynamical methods used in conjunction with those of numerical analysis, spectral theory, the calculus of variations, and many other fields. The reader will find illustrations of this fact in all the papers of the present volume.

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Skew-product semiflows for non-autonomous partial functional differential equations with delay
Sylvia Novo Carmen Núñez Rafael Obaya Ana M. Sanz
Discrete & Continuous Dynamical Systems - A 2014, 34(10): 4291-4321 doi: 10.3934/dcds.2014.34.4291
A detailed dynamical study of the skew-product semiflows induced by families of AFDEs with infinite delay on a Banach space is carried over. Applications are given for families of non-autonomous quasimonotone reaction-diffusion PFDEs with delay in the nonlinear reaction terms, both with finite and infinite delay. In this monotone setting, relations among the classical concepts of sub and super solutions and the dynamical concept of semi-equilibria are established, and some results on the existence of minimal semiflows with a particular dynamical structure are derived.
keywords: infinite delay. skew-product semiflows Topological dynamics finite delay abstract functional differential equations partial functional differential equations
Exponential stability in non-autonomous delayed equations with applications to neural networks
Sylvia Novo Rafael Obaya Ana M. Sanz
Discrete & Continuous Dynamical Systems - A 2007, 18(2&3): 517-536 doi: 10.3934/dcds.2007.18.517
We consider the skew-product semiflow induced by a family of finite-delay functional differential equations and we characterize the exponential stability of its minimal subsets. In the case of non-autonomous systems modelling delayed cellular neural networks, the existence of a global exponentially attracting solution is deduced from the uniform asymptotical stability of the null solution of an associated non-autonomous linear system.
keywords: skew-product semiflows almost periodic solution. global asymptotic stability Non-autonomous dynamical systems neural networks
Principal Floquet subspaces and exponential separations of type Ⅱ with applications to random delay differential equations
Janusz Mierczyński Sylvia Novo Rafael Obaya
Discrete & Continuous Dynamical Systems - A 2018, 38(12): 6163-6193 doi: 10.3934/dcds.2018265

This paper deals with the study of principal Lyapunov exponents, principal Floquet subspaces, and exponential separation for positive random linear dynamical systems in ordered Banach spaces. The main contribution lies in the introduction of a new type of exponential separation, called of type Ⅱ, important for its application to random differential equations with delay. Under weakened assumptions, the existence of an exponential separation of type Ⅱ in an abstract general setting is shown, and an illustration of its application to dynamical systems generated by scalar linear random delay differential equations with finite delay is given.

keywords: Random dynamical systems focusing property generalized principal Floquet bundle generalized exponential separation random delay differential equations

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