Mixed-Type Functional Differential Equations: A $C_{0}$-Semigroup Approach

Year 2018, Volume: 1 Issue: 2, 113 - 125, 24.12.2018

Luis Gerardo Mármol , Carmen Judith Vanegas

https://doi.org/10.33434/cams.446386

Cited By: 1

Abstract

In this paper we study certain systems of mixed-type functional differential equations, from the point of view of the $C_{0}$-semigroup theory. In general, this type of equations are not well-posed as initial value problems. But there are also cases where a unique differentiable solution exists. For these cases and in order to achieve our goal, we first rewrite the system as a classical Cauchy problem in a suitable Banach space. Second, we introduce the associated semigroup and its infinitesimal generator and prove important properties of these operators. As an application, we use the results to characterize the null controllability for those systems, where the control $u$ is constrained to lie in a non-empty compact convex subset $\Om{}$ of $\R^n$.

Keywords

Functional differential equations, Strongly continuous semigroups, Mixed-type difference-differential equations, Exact controllability

References

• [1] A. Rustichini, Functional differential equations of mixed type: the linear autonomous case, J. Dynam. Differ. Equ., 1(2) (1989), 121-143.
• [2] A. Rustichini, Hopf bifurcation of functional differential equations of mixed type, J. Dynam. Differ. Equ., 1(2) (1989), 145-177.
• [3] K. Abell, C. Elmer, A. Humphries, E. Vleck, Computation of mixed type functional differential boundary value problems, SIAM J. Appl. Dyn. Syst., 4(3) (2005), 755–781.
• [4] J. Harterich, B. Sandstede, A. Scheel, Exponential dichotomies for linear non-autonomous functional differential equations of mixed-type, Indiana Univ. Math. J., 51(5) (2002), 94-101.
• [5] N. J. Ford, P. M. Lumb, Mixed-type functional differential equations: a numerical approach, J. Comput. Appl. Math., 229(2) (2009), 471-479.
• [6] N. J. Ford, P. M. Lima, P. M. Lumb, M. F. Teodoro, Numerical approximation of forward-backward differential equations by a finite element method, Proceedings of the International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE 2009), 30 June, 1-3 July (2009).
• [7] N. J. Ford, P. M. Lima, P. M. Lumb, M. F. Teodoro, Numerical modelling of a functional differential equation with deviating arguments using a collocation method, International Conference on Numerical Analysis and Applied Mathematics, (Kos 2008), AIP Proc. 1048 (2008), 553-557.
• [8] V. Iakovleva, C. J. Vanegas, On the solution of differential equations with delayed and advanced arguments, Electron. J. Differ. Equ. Conf., 13 (2005), 57-63.
• [9] V. Iakovleva, R. Manzanilla, L. G. M´armol, C. J. Vanegas, Solutions and constrained-null controllability for a differentialdifference equation, Math. Slovaca, 66(1) (2016), 169-184.
• [10] J. Mallet-Paret, S. M. Verduyn Lunel, Mixed-type functional differential equations, holomorphic factorization and applications, Proc. Equ. Diff. 2003, Inter. Conf. Diff. Equations, (HASSELT 2003), World Scientific, Singapore (2005), 73-89.
• [11] R. F. Curtain, H. J. Zwart, An Introduction to Infinite-dimensional Linear Systems Theory, Texts in Applied Mathematics 21, Springer Verlag, New York-Berlin, 1995.
• [12] A. Carrasco, H. Leiva, Approximate controllability of a system of parabolic equations with delay, J. Math. Anal. Appl., 345 (2008), 845-853.
• [13] R. Manzanilla, L. G. Marmol, C. J. Vanegas, On the controllability of a differential equation with delayed and advanced arguments, Abst. Appl. Anal., 2010, 1-16, Article ID 307409, doi: 10.1155/2010/307409.
• [14] R. F. Brammer, Controllability in linear autonomous systems with positive controllers, SIAM J. Control Optim., 10(2) (1972), 329-353.
• [15] G. Peichl, W. Schappacher, Constrained controllability in Banach spaces, SIAM J. Control Optim., 24 (1986), 1261-1275.
• [16] D. B´arcenas, J. Diestel, Constrained controllability in non reflexive Banach spaces, Quaest. Math., 18 (1995), 185-198.
• [17] J. Diestel, Grotendieck spaces and vector measures, contained in vector and operator valued measures and applications, Proc. Sympos. Alta Utah, 1972, 97-108, Academic Press, NY, USA, 1973.
• [18] J. Diestel, A survey of results related to the Dunford-Pettis property, Sovrem. Mat., AMS, Providence, R.I. USA, 2 (1980).
• [19] H. P. Lotz, Uniform convergence of operators on L¥ and similar spaces, Math. Z., 190 (1985), 207-220.
• [20] D. Barcenas, L. G. M´armol, On the adjoint of as strongly continuous semigroup, Abstr. Appl. Anal., (2008), Article ID 651294.
• [21] H. H. Schaefer, Banach Lattices and Positive Operators, Springer Verlag, 1974.