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2015, 2015(special): 287-296. doi: 10.3934/proc.2015.0287

On the properties of solutions set for measure driven differential inclusions

Mieczysław Cichoń 1, and  Bianca Satco 2,

1. 

Faculty of Mathematics and Computer Science, A. Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland
2. 

Faculty of Electrical Engineering and Computer Science, Stefan cel Mare University, Universitatii 13, 720229 Suceava, Romania

Received  July 2014 Revised  November 2014 Published  November 2015

The aim of the paper is to present properties of solutions set for differential inclusions driven by a positive finite Borel measure. We provide for the most natural type of solution results concerning the continuity of the solution set with respect to the data similar to some already known results, available for different types of solutions. As consequence, the solution set is shown to be compact as a subset of the space of regulated functions. The results allow one (by taking the measure $\mu$ of a particular form) to obtain information on the solution set for continuous or discrete problems, as well as impulsive or retarded set-valued problems.
Keywords: regulated function, Measure differential inclusion, solution set., Lebesgue-Stieltjes integral.
Mathematics Subject Classification: Primary: 34A60, 93C30; Secondary: 28A33, 26A42, 34A3.
Citation: Mieczysław Cichoń, Bianca Satco. On the properties of solutions set for measure driven differential inclusions. Conference Publications, 2015, 2015 (special) : 287-296. doi: 10.3934/proc.2015.0287
References:
[1]

J.P. Aubin and H. Frankowska, "Set-Valued Analysis", Birkhäuser, Boston, 1990.  Google Scholar

[2]

P. Billingsley, Weak convergence of measures: Applications in probability, in "DCBMS-NSF Regional Conference Series in Applied Mathematics", 1971.  Google Scholar

[3]

A.M. Bruckner, J.B. Bruckner and B.S. Thomson, "Real Analysis", Prentice-Hall, 1997. Google Scholar

[4]

C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions", in Lecture Notes in Math. 580, Springer, Berlin, 1977.  Google Scholar

[5]

M. Cichoń and B. Satco, Measure differential inclusions - between continuous and discrete, Adv. Diff. Equations 2014, 2014:56, 18 pp. Google Scholar

[6]

G. Dal Maso and F. Rampazzo, On systems of ordinary differential equations with measures as controls, Differential Integral Equations 4 (1991), 739-765.  Google Scholar

[7]

M. Federson, J.G. Mesquita and A. Slavik, Measure functional differential equations and functional dynamic equations on time scales, J. Diff. Equations 252 (2012), 3816-3847.  Google Scholar

[8]

D. Fraňková, Regulated functions, Math. Bohem. 116 (1991), 20-59.  Google Scholar

[9]

Fremlin, D.H., Measure Theory. Vol. 2, Torres Fremlin, Colchester (2003).  Google Scholar

[10]

Z. Halas and M. Tvrdý, Continuous dependence of solutions of generalized linear differential equations on a parameter, Funct. Differ. Equ. 16 (2009), 299-313.  Google Scholar

[11]

R. Lucchetti, G. Salinetti and R. J-B. Wets, Uniform convergence of probability measures: topological criteria, Jour. Multivariate Anal. 51 (1994), 252-264.  Google Scholar

[12]

J. Lygeros, M. Quincampoix and T. Rze.zuchowski, Impulse differential inclusions driven by discrete measures, in "Hybrid Systems: Computation and Control", Lecture Notes in Computer Science 4416 (2007), 385-398.  Google Scholar

[13]

B. Miller, The generalized solutions of ordinary differential equations in the impulse control problems, J. Math. Syst. Estimation and Control 4 (1994), 1-21.  Google Scholar

[14]

B. Miller and E.Y. Rubinovitch, "Impulsive Control in Continuous and Discrete-Continuous Systems", Kluwer Academic Publishers, Dordrecht, 2003.  Google Scholar

[15]

G. A. Monteiro and M. Tvrdý, Generalized linear differential equations in a Banach space: Continuous dependence on a parameter, Discrete Contin. Dyn. Syst. 33 (2013), 283-303.  Google Scholar

[16]

W.R. Pestman, Measurability of linear operators in the Skorokhod topology, Bull. Belg. Math. Soc. 2 (1995), 381-388.  Google Scholar

[17]

R.R. Rao, Relations between weak and uniform convergence of measures with applications The Annals of Mathematical Statistics 33 (1962), 659-680.  Google Scholar

[18]

S. Saks, "Theory of the Integral", Monografie Matematyczne, Warszawa, 1937. Google Scholar

[19]

Š. Schwabik, M. Tvrdý and O. Vejvoda, "Differential and Integral Equations. Boundary Problems and Adjoints", Dordrecht, Praha, 1979.  Google Scholar

[20]

A.N. Sesekin and S.T. Zavalishchin, "Dynamic Impulse Systems", Dordrecht, Kluwer Academic, 1997.  Google Scholar

[21]

G.N. Silva and R.B. Vinter, Measure driven differential inclusions, J. Math. Anal. Appl. 202 (1996), 727-746.  Google Scholar

[22]

A. Slavik, Well-posedness results for abstract generalized differential equations and measure functional differential equations, Journal of Differential Equations 259 (2015), 666-707. Google Scholar

[23]

M. Tvrdý, "Differential and Integral Equations in the Space of Regulated Functions", Habil. Thesis, Praha, 2001. Google Scholar

show all references

References:
[1]

J.P. Aubin and H. Frankowska, "Set-Valued Analysis", Birkhäuser, Boston, 1990.  Google Scholar

[2]

P. Billingsley, Weak convergence of measures: Applications in probability, in "DCBMS-NSF Regional Conference Series in Applied Mathematics", 1971.  Google Scholar

[3]

A.M. Bruckner, J.B. Bruckner and B.S. Thomson, "Real Analysis", Prentice-Hall, 1997. Google Scholar

[4]

C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions", in Lecture Notes in Math. 580, Springer, Berlin, 1977.  Google Scholar

[5]

M. Cichoń and B. Satco, Measure differential inclusions - between continuous and discrete, Adv. Diff. Equations 2014, 2014:56, 18 pp. Google Scholar

[6]

G. Dal Maso and F. Rampazzo, On systems of ordinary differential equations with measures as controls, Differential Integral Equations 4 (1991), 739-765.  Google Scholar

[7]

M. Federson, J.G. Mesquita and A. Slavik, Measure functional differential equations and functional dynamic equations on time scales, J. Diff. Equations 252 (2012), 3816-3847.  Google Scholar

[8]

D. Fraňková, Regulated functions, Math. Bohem. 116 (1991), 20-59.  Google Scholar

[9]

Fremlin, D.H., Measure Theory. Vol. 2, Torres Fremlin, Colchester (2003).  Google Scholar

[10]

Z. Halas and M. Tvrdý, Continuous dependence of solutions of generalized linear differential equations on a parameter, Funct. Differ. Equ. 16 (2009), 299-313.  Google Scholar

[11]

R. Lucchetti, G. Salinetti and R. J-B. Wets, Uniform convergence of probability measures: topological criteria, Jour. Multivariate Anal. 51 (1994), 252-264.  Google Scholar

[12]

J. Lygeros, M. Quincampoix and T. Rze.zuchowski, Impulse differential inclusions driven by discrete measures, in "Hybrid Systems: Computation and Control", Lecture Notes in Computer Science 4416 (2007), 385-398.  Google Scholar

[13]

B. Miller, The generalized solutions of ordinary differential equations in the impulse control problems, J. Math. Syst. Estimation and Control 4 (1994), 1-21.  Google Scholar

[14]

B. Miller and E.Y. Rubinovitch, "Impulsive Control in Continuous and Discrete-Continuous Systems", Kluwer Academic Publishers, Dordrecht, 2003.  Google Scholar

[15]

G. A. Monteiro and M. Tvrdý, Generalized linear differential equations in a Banach space: Continuous dependence on a parameter, Discrete Contin. Dyn. Syst. 33 (2013), 283-303.  Google Scholar

[16]

W.R. Pestman, Measurability of linear operators in the Skorokhod topology, Bull. Belg. Math. Soc. 2 (1995), 381-388.  Google Scholar

[17]

R.R. Rao, Relations between weak and uniform convergence of measures with applications The Annals of Mathematical Statistics 33 (1962), 659-680.  Google Scholar

[18]

S. Saks, "Theory of the Integral", Monografie Matematyczne, Warszawa, 1937. Google Scholar

[19]

Š. Schwabik, M. Tvrdý and O. Vejvoda, "Differential and Integral Equations. Boundary Problems and Adjoints", Dordrecht, Praha, 1979.  Google Scholar

[20]

A.N. Sesekin and S.T. Zavalishchin, "Dynamic Impulse Systems", Dordrecht, Kluwer Academic, 1997.  Google Scholar

[21]

G.N. Silva and R.B. Vinter, Measure driven differential inclusions, J. Math. Anal. Appl. 202 (1996), 727-746.  Google Scholar

[22]

A. Slavik, Well-posedness results for abstract generalized differential equations and measure functional differential equations, Journal of Differential Equations 259 (2015), 666-707. Google Scholar

[23]

M. Tvrdý, "Differential and Integral Equations in the Space of Regulated Functions", Habil. Thesis, Praha, 2001. Google Scholar

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