# American Institute of Mathematical Sciences

April  2011, 29(2): 467-484. doi: 10.3934/dcds.2011.29.467

## The DuBois-Reymond differential inclusion for autonomous optimal control problems with pointwise-constrained derivatives

 1 Cima-ue, Rua Romão Ramalho 59, P-7000-671 Évora, Portugal

Received  September 2009 Revised  April 2010 Published  October 2010

We prove validity of the classical DuBois-Reymond differential inclusion for the minimizers $y(\cdot)$ of the integral

$\int_{a}^{b}L( x( t) ,x^'( t)) d\,t,\text{ \ }x\( \cdot) \in W^{1,1}((a,b) ,\mathbb{R}^{n}) ,\text{ \ }x(a)=A\,x(b) =B\ \$(*)

whose velocities are not a.e. constrained by the domain boundary.
Thus we do not ask ( as preceding results do) the free-velocity times

$T_{f ree}:=\{ t\in[ a,b] :y^'( t) \in$int$\text{ }dom\ L( y\( t) ,\cdot) \}$

to have "full measure"; on the contrary, "positive measure" of $T_{f ree}$ suffices here to guarantee the above necessary condition.
One main feature of our result is that $L( S,\xi) =\infty$ freely allowed, hence the domains $dom$$L( S,\cdot)$ may be e.g. compact and (*) can be seen as the variational reformulation of general state-and-velocity constrained optimal control problems.
Another main feature is the clean generality of our assumptions on $L( \cdot) :$ any Borel-measurable function $L:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow[ 0,\infty]$ having $L( \cdot,0)$ $lsc$ and $L( S,\cdot)$ convex $lsc$ $\forall\,S.$
The nonconvex case is also considered, for $L( S,\cdot)$ almost convex lsc $\forall\,S.$

Citation: Clara Carlota, António Ornelas. The DuBois-Reymond differential inclusion for autonomous optimal control problems with pointwise-constrained derivatives. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 467-484. doi: 10.3934/dcds.2011.29.467
##### References:
 [1] L. Ambrosio, O. Ascenzi and G. Buttazzo, Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands, J. Math. Anal. Appl., 142 (1989), 301-316. [2] P. Cannarsa, H. Frankowska and E. M. Marchini, On Bolza optimal control problems with constraints, Discr. Cont. Dynam. Syst. Ser. B, 11 (2009), 629-653. doi: doi:10.3934/dcdsb.2009.11.629. [3] C. Carlota, S. Chá and A. Ornelas, Existence of Lipschitz minimizers for nonconvex noncoercive autonomous 1-dim integrals, in preparation. [4] C. Carlota and S. Chá, Existence of Lipschitz optimal arcs and necessary conditions for autonomous Bolza control problems under state and velocity pointwise constraints, in preparation. [5] C. Carlota, Existence of optimal arcs for nonautonomous convex Bolza control problems under pointwise velocity constraints, in preparation. [6] C. Carlota and A. Ornelas, Existence of vector minimizers for nonconvex 1-dim integrals with almost convex Lagrangian, J. Diff. Eqs., 243 (2007), 414-426. [7] A. Cellina and A. Ferriero, Existence of Lipschitzian solutions to the classical problem of the calculus of variations in the autonomous case, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 911-919. doi: doi:10.1016/S0294-1449(03)00010-6. [8] A. Cellina and A. Ornelas, Existence of solutions to differential inclusions and to time optimal control problems in the autonomous case, SIAM J. Control Optim., 42 (2003), 260-265. doi: doi:10.1137/S0363012902408046. [9] L. Cesari, "Optimization, Theory and Applications," Springer, Berlin, 1983. [10] F. H. Clarke, An indirect method in the calculus of variations, Trans. Amer. Math. Soc., 336 (1993), 655-673. doi: doi:10.2307/2154369. [11] F. H. Clarke, Necessary conditions in dynamic optimization, Mem. Amer. Math. Soc., 173 (2005), x+113. [12] F. H. Clarke, "Necessary Conditions in Optimal Control and in the Calculus of Variations, Differential Equations, Chaos and Variational Problems," Prog. in Nonlinear Diff. Eq. Appl., 75, Birkhäuser, Basel, (2008), 143-156. [13] G. Dal Maso and H. Frankowska, Autonomous integral functionals with discontinuous nonconvex integrands: Lipschitz regularity of minimizers, DuBois-Reymond necessary conditions, and Hamilton-Jacobi equations, Appl. Math. Optim., 48 (2003), 39-66. doi: doi:10.1007/s00245-003-0768-4. [14] C. Olech, Existence theory in optimal control problems - the underlying ideas, "International Conference on Differential Equations" (Proc., Univ. Southern California, Los Angeles, Calif., 1974), Academic Press, New York, (1975), 612-635. [15] A. Ornelas, Existence of scalar minimizers for simple convex integrals with autonomous Lagrangian measurable on the state variable, Nonlinear Anal., 67 (2007), 2485-2496. doi: doi:10.1016/j.na.2006.08.044. [16] R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 1970. [17] R. T. Rockafellar, Existence and duality theorems for convex problems of Bolza, Trans. Amer. Math. Soc., 159 (1971), 1-40. [18] R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis," Springer-Verlag, Berlin, 1997. [19] R. B. Vinter, "Optimal Control," Birkhäuser, Berlin, 2000.

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##### References:
 [1] L. Ambrosio, O. Ascenzi and G. Buttazzo, Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands, J. Math. Anal. Appl., 142 (1989), 301-316. [2] P. Cannarsa, H. Frankowska and E. M. Marchini, On Bolza optimal control problems with constraints, Discr. Cont. Dynam. Syst. Ser. B, 11 (2009), 629-653. doi: doi:10.3934/dcdsb.2009.11.629. [3] C. Carlota, S. Chá and A. Ornelas, Existence of Lipschitz minimizers for nonconvex noncoercive autonomous 1-dim integrals, in preparation. [4] C. Carlota and S. Chá, Existence of Lipschitz optimal arcs and necessary conditions for autonomous Bolza control problems under state and velocity pointwise constraints, in preparation. [5] C. Carlota, Existence of optimal arcs for nonautonomous convex Bolza control problems under pointwise velocity constraints, in preparation. [6] C. Carlota and A. Ornelas, Existence of vector minimizers for nonconvex 1-dim integrals with almost convex Lagrangian, J. Diff. Eqs., 243 (2007), 414-426. [7] A. Cellina and A. Ferriero, Existence of Lipschitzian solutions to the classical problem of the calculus of variations in the autonomous case, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 911-919. doi: doi:10.1016/S0294-1449(03)00010-6. [8] A. Cellina and A. Ornelas, Existence of solutions to differential inclusions and to time optimal control problems in the autonomous case, SIAM J. Control Optim., 42 (2003), 260-265. doi: doi:10.1137/S0363012902408046. [9] L. Cesari, "Optimization, Theory and Applications," Springer, Berlin, 1983. [10] F. H. Clarke, An indirect method in the calculus of variations, Trans. Amer. Math. Soc., 336 (1993), 655-673. doi: doi:10.2307/2154369. [11] F. H. Clarke, Necessary conditions in dynamic optimization, Mem. Amer. Math. Soc., 173 (2005), x+113. [12] F. H. Clarke, "Necessary Conditions in Optimal Control and in the Calculus of Variations, Differential Equations, Chaos and Variational Problems," Prog. in Nonlinear Diff. Eq. Appl., 75, Birkhäuser, Basel, (2008), 143-156. [13] G. Dal Maso and H. Frankowska, Autonomous integral functionals with discontinuous nonconvex integrands: Lipschitz regularity of minimizers, DuBois-Reymond necessary conditions, and Hamilton-Jacobi equations, Appl. Math. Optim., 48 (2003), 39-66. doi: doi:10.1007/s00245-003-0768-4. [14] C. Olech, Existence theory in optimal control problems - the underlying ideas, "International Conference on Differential Equations" (Proc., Univ. Southern California, Los Angeles, Calif., 1974), Academic Press, New York, (1975), 612-635. [15] A. Ornelas, Existence of scalar minimizers for simple convex integrals with autonomous Lagrangian measurable on the state variable, Nonlinear Anal., 67 (2007), 2485-2496. doi: doi:10.1016/j.na.2006.08.044. [16] R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 1970. [17] R. T. Rockafellar, Existence and duality theorems for convex problems of Bolza, Trans. Amer. Math. Soc., 159 (1971), 1-40. [18] R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis," Springer-Verlag, Berlin, 1997. [19] R. B. Vinter, "Optimal Control," Birkhäuser, Berlin, 2000.
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