# American Institute of Mathematical Sciences

December 2018, 11(6): 1201-1218. doi: 10.3934/dcdss.2018068

## On ${\mathcal L}^1$ limit solutions in impulsive control

 Dipartimento di Matematica "Tullio Levi-Civita", Università di Padova, Via Trieste, 63, Padova 35121, Italy

* Corresponding author: Monica Motta

Received  June 2017 Revised  October 2017 Published  June 2018

Fund Project: This research is partially supported by the INdAM-GNAMPA Project 2017 "Optimal impulsive control: higher order necessary conditions and gap phenomena"; and by the Padova University grant PRAT 2015 "Control of dynamics with reactive constraints"

We consider a nonlinear control system depending on two controls $u$ and $v$, with dynamics affine in the (unbounded) derivative of $u$, and $v$ appearing initially only in the drift term. Recently, motivated by applications to optimization problems lacking coercivity, Aronna and Rampazzo [1] proposed a notion of generalized solution $x$ for this system, called limit solution, associated to measurable $u$ and $v$, and with $u$ of possibly unbounded variation in $[0, T]$. As shown in [1], when $u$ and $x$ have bounded variation, such a solution (called in this case BV simple limit solution) coincides with the most used graph completion solution (see e.g. Rishel [25], Warga [27] and Bressan and Rampazzo [8]). In [24] we extended this correspondence to BV$_{loc}$ inputs $u$ and trajectories (with bounded variation just on any $[0, t]$ with $t<T$). Here, starting with an example of optimal control where the minimum does not exist in the class of limit solutions, we propose a notion of extended limit solution $x$, for which such a minimum exists. As a first result, we prove that extended BV (respectively, BV$_{loc}$) simple limit solutions and BV (respectively, BV$_{loc}$) simple limit solutions coincide. Then we consider dynamics where the ordinary control $v$ also appears in the non-drift terms. For the associated system we prove that, in the BV case, extended limit solutions coincide with graph completion solutions.

Citation: Monica Motta, Caterina Sartori. On ${\mathcal L}^1$ limit solutions in impulsive control. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1201-1218. doi: 10.3934/dcdss.2018068
##### References:
 [1] M. S. Aronna and F. Rampazzo, ${\mathcal L}^1$ limit solutions for control systems, J. Differential Equations, 258 (2015), 954-979. doi: 10.1016/j.jde.2014.10.013. [2] M. S. Aronna, M. Motta and F. Rampazzo, Infimum gaps for limit solutions, Set-Valued Var. Anal., 23 (2015), 3-22. doi: 10.1007/s11228-014-0296-1. [3] A. Arutyunov, D. Karamzin and F. Pereira, On a generalization of the impulsive control concept: Controlling system jumps, Discrete Contin. Dyn. Syst., 29 (2011), 403-415. doi: 10.3934/dcds.2011.29.403. [4] J.-P. Aubin, Impulse Differential Equations and Hybrid Systems: A Viability Approach, Lecture Notes. University of California, Berkeley, 2000. [5] E. Barron and H. Ishii, The Bellman equation for minimizing the maximum cost, Nonlinear Anal., 13 (1989), 1067-1090. doi: 10.1016/0362-546X(89)90096-5. [6] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series on Applied Mathematics, 2. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007. [7] A. Bressan and F. Rampazzo, Moving constraints as stabilizing controls in classical mechanics, Arch. Ration. Mech. Anal., 196 (2010), 97-141. doi: 10.1007/s00205-009-0237-6. [8] A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital. B, 2 (1988), 641-656. [9] A. Bressan and F. Rampazzo, Impulsive control systems with commutative vector fields, J. Optim. Theory Appl., 71 (1991), 67-83. doi: 10.1007/BF00940040. [10] A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J. Optim. Theory Appl., 81 (1994), 435-457. doi: 10.1007/BF02193094. [11] N. Falkner and G. Teschl, On the substitution rule for Lebesgue-Stieltjes integrals, Expo. Math., 30 (2012), 412-418. doi: 10.1016/j.exmath.2012.09.002. [12] S. L. Fraga, R. Gomes and F. L. Pereira, An impulsive framework for the control of hybrid systems, in Proc. 46 IEEE Conf. Decision Control, (2007), 5444-5449. doi: 10.1109/CDC.2007.4434895. [13] E. Goncharova and M. Staritsyn, Optimization of Measure-Driven Hybrid Systems, J. Optim. Theory Appl., 153 (2012), 139-156. doi: 10.1007/s10957-011-9944-x. [14] M. Guerra and A. Sarychev, Fréchet generalized trajectories and minimizers for variational problems of low coercivity, J. Dyn. Control Syst., 21 (2015), 351-377. doi: 10.1007/s10883-014-9231-x. [15] W. Haddad, V. Chellaboina and S. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton University Press, Princeton, 2006. doi: 10.1515/9781400865246. [16] D. Y. Karamzin, V. A. de Oliveira, F. L. Pereira and G. N. Silva, On the properness of an impulsive control extension of dynamic optimization problems, ESAIM Control Optim. Calc. Var., 21 (2015), 857-875. doi: 10.1051/cocv/2014053. [17] A. Kurzhanski and P. Tochilin, Impulse controls in models of hybrid systems, Differential Equations, 45 (2009), 731-742. doi: 10.1134/S0012266109050127. [18] T. Lyons and Z. Qian, System Control and Rough Paths, Oxford Mathematical Monographs. Oxford Science Publications. Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198506485.001.0001. [19] B. Miller and E. Y. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic/Plenum Publishers, New York, 2003. doi: 10.1007/978-1-4615-0095-7. [20] M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential Integral Equations, 8 (1995), 269-288. [21] M. Motta and F. Rampazzo, Dynamic programming for nonlinear systems driven by ordinary and impulsive controls, SIAM J. Control Optim., 34 (1996), 199-225. doi: 10.1137/S036301299325493X. [22] M. Motta and C. Sartori, Uniqueness results for boundary value problems arising from finite fuel and other singular and unbounded stochastic control problems, Discrete Contin. Dyn. Syst., 21 (2008), 513-535. doi: 10.3934/dcds.2008.21.513. [23] M. Motta and C. Sartori, On asymptotic exit-time control problems lacking coercivity, ESAIM Control Optim. Calc. Var., 20 (2014), 957-982. doi: 10.1051/cocv/2014003. [24] M. Motta and C. Sartori, Lack of BV bounds for impulsive control systems, J. Math. Anal. Appl., 461 (2018), 422-450. [25] R. W. Rishel, An extended Pontryagin principle for control systems whose control laws contain measures, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 191-205. doi: 10.1137/0303016. [26] G. Silva and R. Vinter, Measure driven differential inclusions, J. Math. Anal. Appl., 202 (1996), 727-746. doi: 10.1006/jmaa.1996.0344. [27] J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972. [28] H. Whitney, Functions differentiable on the boundaries of regions, Ann. of Math., 35 (1934), 482-485. doi: 10.2307/1968745. [29] P. Wolenski and S. Žabić, A sampling method and approximation results for impulsive systems, SIAM J. Control Optim., 46 (2007), 983-998. doi: 10.1137/040620734. [30] K. Yunt, Modelling of mechanical blocking, Recent Researches in Circuits, Systems, Mechanics and Transportation Systems, (2011), 123-128.

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##### References:
 [1] M. S. Aronna and F. Rampazzo, ${\mathcal L}^1$ limit solutions for control systems, J. Differential Equations, 258 (2015), 954-979. doi: 10.1016/j.jde.2014.10.013. [2] M. S. Aronna, M. Motta and F. Rampazzo, Infimum gaps for limit solutions, Set-Valued Var. Anal., 23 (2015), 3-22. doi: 10.1007/s11228-014-0296-1. [3] A. Arutyunov, D. Karamzin and F. Pereira, On a generalization of the impulsive control concept: Controlling system jumps, Discrete Contin. Dyn. Syst., 29 (2011), 403-415. doi: 10.3934/dcds.2011.29.403. [4] J.-P. Aubin, Impulse Differential Equations and Hybrid Systems: A Viability Approach, Lecture Notes. University of California, Berkeley, 2000. [5] E. Barron and H. Ishii, The Bellman equation for minimizing the maximum cost, Nonlinear Anal., 13 (1989), 1067-1090. doi: 10.1016/0362-546X(89)90096-5. [6] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series on Applied Mathematics, 2. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007. [7] A. Bressan and F. Rampazzo, Moving constraints as stabilizing controls in classical mechanics, Arch. Ration. Mech. Anal., 196 (2010), 97-141. doi: 10.1007/s00205-009-0237-6. [8] A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital. B, 2 (1988), 641-656. [9] A. Bressan and F. Rampazzo, Impulsive control systems with commutative vector fields, J. Optim. Theory Appl., 71 (1991), 67-83. doi: 10.1007/BF00940040. [10] A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J. Optim. Theory Appl., 81 (1994), 435-457. doi: 10.1007/BF02193094. [11] N. Falkner and G. Teschl, On the substitution rule for Lebesgue-Stieltjes integrals, Expo. Math., 30 (2012), 412-418. doi: 10.1016/j.exmath.2012.09.002. [12] S. L. Fraga, R. Gomes and F. L. Pereira, An impulsive framework for the control of hybrid systems, in Proc. 46 IEEE Conf. Decision Control, (2007), 5444-5449. doi: 10.1109/CDC.2007.4434895. [13] E. Goncharova and M. Staritsyn, Optimization of Measure-Driven Hybrid Systems, J. Optim. Theory Appl., 153 (2012), 139-156. doi: 10.1007/s10957-011-9944-x. [14] M. Guerra and A. Sarychev, Fréchet generalized trajectories and minimizers for variational problems of low coercivity, J. Dyn. Control Syst., 21 (2015), 351-377. doi: 10.1007/s10883-014-9231-x. [15] W. Haddad, V. Chellaboina and S. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton University Press, Princeton, 2006. doi: 10.1515/9781400865246. [16] D. Y. Karamzin, V. A. de Oliveira, F. L. Pereira and G. N. Silva, On the properness of an impulsive control extension of dynamic optimization problems, ESAIM Control Optim. Calc. Var., 21 (2015), 857-875. doi: 10.1051/cocv/2014053. [17] A. Kurzhanski and P. Tochilin, Impulse controls in models of hybrid systems, Differential Equations, 45 (2009), 731-742. doi: 10.1134/S0012266109050127. [18] T. Lyons and Z. Qian, System Control and Rough Paths, Oxford Mathematical Monographs. Oxford Science Publications. Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198506485.001.0001. [19] B. Miller and E. Y. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic/Plenum Publishers, New York, 2003. doi: 10.1007/978-1-4615-0095-7. [20] M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential Integral Equations, 8 (1995), 269-288. [21] M. Motta and F. Rampazzo, Dynamic programming for nonlinear systems driven by ordinary and impulsive controls, SIAM J. Control Optim., 34 (1996), 199-225. doi: 10.1137/S036301299325493X. [22] M. Motta and C. Sartori, Uniqueness results for boundary value problems arising from finite fuel and other singular and unbounded stochastic control problems, Discrete Contin. Dyn. Syst., 21 (2008), 513-535. doi: 10.3934/dcds.2008.21.513. [23] M. Motta and C. Sartori, On asymptotic exit-time control problems lacking coercivity, ESAIM Control Optim. Calc. Var., 20 (2014), 957-982. doi: 10.1051/cocv/2014003. [24] M. Motta and C. Sartori, Lack of BV bounds for impulsive control systems, J. Math. Anal. Appl., 461 (2018), 422-450. [25] R. W. Rishel, An extended Pontryagin principle for control systems whose control laws contain measures, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 191-205. doi: 10.1137/0303016. [26] G. Silva and R. Vinter, Measure driven differential inclusions, J. Math. Anal. Appl., 202 (1996), 727-746. doi: 10.1006/jmaa.1996.0344. [27] J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972. [28] H. Whitney, Functions differentiable on the boundaries of regions, Ann. of Math., 35 (1934), 482-485. doi: 10.2307/1968745. [29] P. Wolenski and S. Žabić, A sampling method and approximation results for impulsive systems, SIAM J. Control Optim., 46 (2007), 983-998. doi: 10.1137/040620734. [30] K. Yunt, Modelling of mechanical blocking, Recent Researches in Circuits, Systems, Mechanics and Transportation Systems, (2011), 123-128.
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