# American Institute of Mathematical Sciences

April  2019, 39(4): 2255-2283. doi: 10.3934/dcds.2019095

## On smoothness of solutions to projected differential equations

 1 Lab. PROMES UPR CNRS 8521, Université de Perpignan Via Domitia, Rambla de la Thermodynamique, Tecnosud, F- 66100 Perpignan, France 2 Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, 163 rue Auguste Broussonnet, 34090 Montpellier, France 3 Instituto de Ciencias de la Educación, Universidad de O'Higgins, Av. Libertador Bernardo O'Higgins 611, Rancagua, Chile

* Corresponding author: David Salas

Received  August 2018 Revised  October 2018 Published  January 2019

Projected differential equations are known as fundamental mathematical models in economics, for electric circuits, etc. The present paper studies the (higher order) derivability as well as a generalized type of derivability of solutions of such equations when the set involved for projections is prox-regular with smooth boundary.

Citation: David Salas, Lionel Thibault, Emilio Vilches. On smoothness of solutions to projected differential equations. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2255-2283. doi: 10.3934/dcds.2019095
##### References:

show all references

##### References:
Two prox-regular sets with smooth and non-smooth boundary
Sets of Definition 4.1 for a trajectory in $\mathbb{R} ^2$
A circuit with an ideal diode, an inductor and a current source
Functions $f_1$ and $f_2$
Trajectory for $f_2$
 [1] José A. Carrillo, Dejan Slepčev, Lijiang Wu. Nonlocal-interaction equations on uniformly prox-regular sets. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1209-1247. doi: 10.3934/dcds.2016.36.1209 [2] Pablo Amster, Alberto Déboli, Manuel Pinto. Hartman and Nirenberg type results for systems of delay differential equations under $(\omega,Q)$-periodic conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021171 [3] Christian Aarset, Christian Pötzsche. Bifurcations in periodic integrodifference equations in $C(\Omega)$ Ⅱ: Discrete torus bifurcations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1847-1874. doi: 10.3934/cpaa.2020081 [4] Michal Fečkan, Kui Liu, JinRong Wang. $(\omega,\mathbb{T})$-periodic solutions of impulsive evolution equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021006 [5] Christian Aarset, Christian Pötzsche. Bifurcations in periodic integrodifference equations in $C(\Omega)$ I: Analytical results and applications. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 1-60. doi: 10.3934/dcdsb.2020231 [6] Pallavi Bedi, Anoop Kumar, Thabet Abdeljawad, Aziz Khan. S-asymptotically $\omega$-periodic mild solutions and stability analysis of Hilfer fractional evolution equations. Evolution Equations & Control Theory, 2021, 10 (4) : 733-748. doi: 10.3934/eect.2020089 [7] Yu-Zhao Wang. $\mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116 [8] Jingwen Wu, Jintao Hu, Hongjiong Tian. Functionally-fitted block $\theta$-methods for ordinary differential equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2603-2617. doi: 10.3934/dcdss.2020164 [9] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3351-3386. doi: 10.3934/dcdss.2020440 [10] Luisa Malaguti, Stefania Perrotta, Valentina Taddei. $L^p$-exact controllability of partial differential equations with nonlocal terms. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021053 [11] Emma D'Aniello, Saber Elaydi. The structure of $\omega$-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195 [12] Olexiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Chain recurrence and structure of $\omega$-limit sets of multivalued semiflows. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2197-2217. doi: 10.3934/cpaa.2020096 [13] Vladimir Chepyzhov, Alexei Ilyin, Sergey Zelik. Strong trajectory and global $\mathbf{W^{1,p}}$-attractors for the damped-driven Euler system in $\mathbb R^2$. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1835-1855. doi: 10.3934/dcdsb.2017109 [14] Fuzhi Li, Dongmei Xu, Jiali Yu. Regular measurable backward compact random attractor for $g$-Navier-Stokes equation. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3137-3157. doi: 10.3934/cpaa.2020136 [15] Chunming Tang, Maozhi Xu, Yanfeng Qi, Mingshuo Zhou. A new class of $p$-ary regular bent functions. Advances in Mathematics of Communications, 2021, 15 (1) : 55-64. doi: 10.3934/amc.2020042 [16] Mathew Gluck. Classification of solutions to a system of $n^{\rm th}$ order equations on $\mathbb R^n$. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246 [17] Abdeladim El Akri, Lahcen Maniar. Uniform indirect boundary controllability of semi-discrete $1$-$d$ coupled wave equations. Mathematical Control & Related Fields, 2020, 10 (4) : 669-698. doi: 10.3934/mcrf.2020015 [18] Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2751-2771. doi: 10.3934/cpaa.2018130 [19] Wensheng Yin, Jinde Cao, Guoqiang Zheng. Further results on stabilization of stochastic differential equations with delayed feedback control under $G$-expectation framework. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021072 [20] Thomas French. Follower, predecessor, and extender set sequences of $\beta$-shifts. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4331-4344. doi: 10.3934/dcds.2019175

2020 Impact Factor: 1.392