On the attainable set for scalar balance laws with distributed control

Corghi, Marco; Marson, Andrea

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 236-266.

The paper deals with the set of attainable profiles of a solution $u$ to a scalar balance law in one space dimension with strictly convex flux function

        $\partial {}_{t}u+\partial {}_{x}f\left(u\right)=z(t,x).$

Here the function $z$ is regarded as a bounded measurable control. We are interested in studying the set of attainable profiles at a fixed time $T>0$, both in case $z(t,·)$ is supported in the all real line, and in case $z(t,·)$ is supported in a compact interval $[a,b]$ independent on the time variable $t$.

D. Amadori, L. Gosse and G. Guerra, Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws. Arch. Ration. Mech. Anal. 162 (2002) 327–366. | DOI | Zbl

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000). | Zbl

F. Ancona and G.M. Coclite, On the attainable set for Temple class systems with boundary controls. SIAM J. Control Optim. 43 (2005) 2166–2190. | DOI | Zbl

F. Ancona and P. Goatin, Uniqueness and stability of $L^{\infty }$ solutions for Temple class systems with boundary and properties of the attainable sets. SIAM J. Math. Anal. 34 (2002) 28–63. | DOI | Zbl

F. Ancona and A. Marson, On the attainable set for scalar nonlinear conservation laws with boundary control. SIAM J. Control Optim. 36 (1998) 290–312. | DOI | Zbl

F. Ancona and A. Marson, Asymptotic Stabilization of Systems of Conservation Laws by Controls Acting at a Single Boundary Point, in Control methods in PDE-dynamical systems. Vol. 426 of Contemp. Math. American Mathematical Society, Providence, RI (2007) 1–43 | Zbl

A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem. Oxford University Press, Oxford (2000). | Zbl

A. Bressan and G.M. Coclite, On the boundary control of systems of conservation laws. SIAM J. Control Optim. 41 (2002) 607–622. | DOI | Zbl

J.M. Coron, Control and Nonlinearity. Vol. 136 of Math. Surv. Monogr. American Mathematical Society, Providence, RI (2007). | Zbl

J.M. Coron, O. Glass and Z. Wang, Exact boundary controllability for 1-D quasilinear hyperbolic systems with a vanishing characteristic speed. SIAM J. Control Optim. 48 (2009/10) 3105–3122. | DOI | Zbl

M. Chapouly, Global controllability of nonviscous and viscous Burgers-type equations. SIAM J. Control Optim. 48 (2009) 1567–1599. | DOI | Zbl

C.M. Dafermos, Generalized characteristic and the structure of solutions of hyperbolic conservation laws. Indiana Math. J. 26 (1977) 1097–1119. | DOI | Zbl

C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 2nd edition. Springer-Verlag, Berlin (2005). | Zbl

E. Fernández-Cara and S. Guerrero, Null controllability of the Burgers system with distributed controls. Systems Control Lett. 56 (2007) 366–372. | DOI | Zbl

O. Glass, On the controllability of the 1-D isentropic Euler equation. J. Eur. Math. Soc. 9 (2007) 427–486. | DOI | Zbl

O. Glass, On the controllability of the non-isentropic $1$-D Euler equation. J. Differ. Equ. 257 (2014) 638–719. | DOI | Zbl

T. Horsin, On the controllability of the Burgers equation, ESAIM: COCV 3 (1998) 83–95. | Numdam | MR | Zbl

H. Holden and N.H. Risebro, Front Tracking for Hyperbolic Conservation Laws. Springer-Verlag, New York (2002). | MR | Zbl

S.N. Kruzkov, First order quasilinear equations in several independent variables. Math. USSR Sbornik 10 (1970) 217–243. | DOI | MR | Zbl

M. Léautaud, Uniform controllability of scalar conservation laws in the vanishing viscosity limit. SIAM J. Control Optim. 50 (2012) 1661–1699. | DOI | MR | Zbl

V. Perrollaz, Exact controllability of scalar conservation laws with an additional control in the context of entropy solutions. SIAM J. Control Optim. 50 (2012) 2025–2045. | DOI | MR | Zbl

V. Perrollaz, Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 30 (2013) 879–915. | DOI | Numdam | MR | Zbl

R. Robyr, SBV regularity of entropy solutions for a class of genuinely nonlinear scalar balance laws with non-convex flux function. J. Hyperbolic Differ. Equ. 5 (2008) 449–475. | DOI | MR | Zbl