Global Controllability for Quasilinear Nonnegative Definite System of ODEs and SDEs

We consider exact and averaged control problem for a system of quasi-linear ODEs and SDEs with a non-negative definite symmetric matrix of the system. The strategy of the proof is the standard linearization of the system by fixing the function appearing in the nonlinear part of the system, and then applying the Leray-Schauder fixed point theorem. We shall also need the continuous induction arguments to prolong the control to the final state which is a novel approach in the field. This enables us to obtain controllability for arbitrarily large initial data (so called global controllability). MSC (2010): Primary: 34H05; Secondary: 49J15, 93C15, 60H10

[1]  J. Bismut.  Conjugate convex functions in optimal stochastic control , 1973 .

[2]  Enrique Zuazua,et al.  From averaged to simultaneous controllability of parameter dependent finite-dimensional systems , 2015 .

[3]  Bujar Gashi,et al.  Stochastic minimum-energy control , 2014, Syst. Control. Lett..

[4]  F. Udwadia.  Optimal tracking control of nonlinear dynamical systems , 2008, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[5]  Xu Zhang.  Remarks on the Controllability of Some Quasilinear Equations , 2009, 0904.2427.

[6]  Koichiro Naito,et al.  Controllability of semilinear control systems dominated by the linear part , 1987 .

[7]  Yanqing Wang,et al.  THE NORM OPTIMAL CONTROL PROBLEM FOR STOCHASTIC LINEAR CONTROL SYSTEMS , 2015 .

[8]  J. Vázquez.  The Porous Medium Equation: Mathematical Theory , 2006 .

[9]  Firdaus E. Udwadia,et al.  Explicit solution to the full nonlinear problem for satellite formation-keeping , 2010 .

[10]  Xu Liu,et al.  Local Controllability of Multidimensional Quasi-Linear Parabolic Equations , 2012, SIAM J. Control. Optim..

[11]  S. Peng,et al.  Adapted solution of a backward stochastic differential equation , 1990 .

[12]  Enrique Zuazua,et al.  Averaged control and observation of parameter-depending wave equations , 2014 .

[13]  J. Yong,et al.  Exact Controllability of Linear Stochastic Differential Equations and Related Problems , 2016, 1603.07789.

[14]  Miljana Jovanovic,et al.  Perturbed backward stochastic differential equations , 2012, Math. Comput. Model..

[15]  J. Vázquez.  The Porous Medium Equation , 2006 .

[16]  F. Quirós,et al.  A P ] 1 4 Ja n 20 10 A fractional porous medium equation by , 2010 .

[17]  Enrique Zuazua,et al.  Averaged control , 2014, Autom..

[18]  Jerzy Klamka,et al.  Schauder’s fixed-point theorem in approximate controllability problems , 2016, Int. J. Appl. Math. Comput. Sci..

[19]  Svetlana Jankovic,et al.  Backward stochastic Volterra integral equations with additive perturbations , 2015, Appl. Math. Comput..

[20]  Benjamin Gess,et al.  Finite Speed of Propagation for Stochastic Porous Media Equations , 2012, SIAM J. Math. Anal..

[21]  L. Hörmander,et al.  Lectures on Nonlinear Hyperbolic Differential Equations , 1997 .

[22]  X. Mao,et al.  Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients , 1995 .

[23]  MA BYJIN,et al.  REPRESENTATION THEOREMS FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS , 2002 .

[24]  Jasmina Djordjevic,et al.  Reflected backward stochastic differential equations with perturbations , 2018 .

[25]  R. Kalaba,et al.  A new perspective on constrained motion , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[26]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[27]  Koichiro Naito.  On controllability for a nonlinear Volterra equation , 1992 .

[28]  J. Y. Park,et al.  Approximate controllability for trajectories of a delay Volterra control system , 1989 .

[29]  Enrique Zuazua,et al.  Greedy controllability of finite dimensional linear systems , 2016, Autom..