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In this article we study, by the vanishing viscosity method, the sensitivity analysis of an optimal control problem for 1-D scalar conservation laws in the presence of shocks. It is reduced to investigate the vanishing viscosity limit for the nonlinear conservation law, the corresponding linearized equation and its adjoint equation, respectively. We employ the method of matched asymptotic expansions to construct approximate solutions to those equations. It is then proved that the approximate solutions, respectively, satisfy those viscous equations in the asymptotic sense, and converge to the solutions of the corresponding inviscid problems with certain convergent rates. A new equation for the variation of shock positions is derived. It is also discussed how to identify descent directions to find the minimizer of the viscous optimal control problem in the quasi-shock case.

Optimal control for hyperbolic conservation laws requires a considerable analytical effort and computational expense in practice, is thus a difficult topic. Some methods have been developed in the last several years to reduce the computational cost and to render this type of problems affordable. In particular, recently in [1] Castro, Palacios and Zuazua have developed an “alternating descent method” that takes the possible shock discontinuities into account, for the optimal control of the inviscid Burgers equation in one space dimension. Further in [2] this numerical method is also employed to study the optimal control problem of the Burgers equation with small viscosity via the Hopf–Cole formula which can be found in [3] and [4], for instance. In the present article, we study the sensitivity analysis of an optimal control problem for 1-D general nonlinear scalar conservation law in the presence of shocks by the method of vanishing viscosity. It reduces to study the vanishing viscosity limit of solutions to the nonlinear conservation law, the corresponding linearized equation and its adjoint equation, which is the main result of this paper (see Theorem 4.1). Observing that the discontinuities in those equations will lead to difficulties when passing to the limit, we apply the method of matched asymptotic expansions to study the convergence. In particular, the solutions to the viscous adjoint problem approach a constant in a “triangular region” formed by the characteristics intersecting at the shock, as the viscosity vanishes. Thus it extends the results in [2] for the Burgers equation to a general case. Moreover, certain convergent rates are obtained so that it generalizes the result of James and Sepúlveda (see [5]) for adjoint equations. As a result, it is reasonable to apply the efficient numerical method, such as the “alternating descent method” developed in [2], to study the optimal control problem for general conservation laws with small viscosity, when the solutions to the corresponding hyperbolic conservation laws have shock discontinuities. Another main feature of this article is that, we have derived a new equation for the variation of shock position, which approaches, as a parameter tend to infinity, to the one obtained by Bressan and Marson (see [6]). We first state the optimal control problem as follows. For a given T>0T>0, we study the following inviscid problem equation(1.1) View the MathML sourceut+(F(u))x=0,inR×(0,T), Turn MathJax on equation(1.2) View the MathML sourceu(x,0)=uI(x),x∈R, Turn MathJax on where F:R→RF:R→R is a smooth function, f=Fuf=Fu and uIuI is a piecewise smooth function. A function u(x,t)u(x,t) is called a single-shock solution (see e.g., [7]) to (1.1)– (1.2) up to time TT if 1. u(x,t)u(x,t) is a distributional solution of the hyperbolic system (1.1)–(1.2) in the region R×[0,T]R×[0,T]. 2. There exists a smooth curve, the shock, View the MathML sourcex=φ(t),0≤t≤T, so that u(x,t)u(x,t) is sufficiently smooth at any point x≠φ(t)x≠φ(t). 3. The limits View the MathML source∂xku(φ(t)−0,t)=limx→φ(t)−0∂xku(x,t),∂xku(φ(t)+0,t)=limx→φ(t)+0∂xku(x,t), Turn MathJax on exist and are finite for t≤Tt≤T and k=0,1,2,3,4k=0,1,2,3,4. 4. The entropy condition is satisfied at x=φ(t)x=φ(t), that is, f(u(φ(t)−0,t))>φ′(t)>f(u(φ(t)+0,t)).f(u(φ(t)−0,t))>φ′(t)>f(u(φ(t)+0,t)). Turn MathJax on Although we present the results for single-shock solutions, it will be clear from our analysis that similar results hold for piece-wise smooth solutions with finitely many non-interacting shocks. Given a target function uD∈L2(R)uD∈L2(R) we consider the cost functional to be minimized J:L1(R)→RJ:L1(R)→R, defined by equation(1.3) J(uI)=∫R|u(x,T)−uD(x)|2dx,J(uI)=∫R|u(x,T)−uD(x)|2dx, Turn MathJax on where u(x,t)u(x,t) is the unique single-shock solution to (1.1)– (1.2). We also introduce the set of admissible initial data Uad⊂L1(R)Uad⊂L1(R), that we shall define later in order to guarantee the existence of the following optimization problem: Find uI,min∈UaduI,min∈Uad such that View the MathML sourceJ(uI,min)=minuI∈UadJ(uI). Turn MathJax on Such a problem has been studied in e.g. [1] and [2] in the case that View the MathML sourceF(u)=u22. This is one of the model optimization problems that is often addressed in the context of optimal aerodynamic design (see [8]). In practical applications, in order to perform numerical computations and simulations one has to replace the continuous optimization problem above by a discrete approximation, and develop efficient algorithms for computing accurate approximations of the discrete minimizers. The most efficient methods to approximate minimizers are the gradient methods: one first linearizes (1.1) to obtain a descent direction of the continuous functional JJ, then takes a numerical approximation of this descent direction with the discrete values provided by the numerical scheme. To this end, we first linearize (1.1) to yield equation(1.4) View the MathML source∂t(δu)+∂x(f(u)δu)=0,inR×(0,T), Turn MathJax on where δuδu is the variation of uu with respect to uIuI. However, it is not justified when the solutions have shocks since singular terms may appear on the linearization over the shock location. Accordingly in optimal control applications we also need to take into account the sensitivity for the shock location (which has been studied by many authors, see, e.g. [1], [2], [6], [9] and [10]). Roughly speaking, the main conclusion of that analysis is that the classical linearized equation (1.4) must be complemented with an equation for the sensitivity of the shock positions (see Section 3.2). Moreover, it is also necessary to carry out the sensitivity analysis of the optimal control problem for the numerical purpose. As we shall see in Section 2.3, it reduces to study the following (inverse) adjoint problem corresponding to (1.4), equation(1.5) View the MathML source−pt−f(u)px=0,inR×(0,T), Turn MathJax on equation(1.6) View the MathML sourcep(x,T)=pT(x)≡u(x,T)−uD(x),x∈R. Turn MathJax on Next, we introduce accordingly the viscous problem corresponding to (1.1) as equation(1.7) View the MathML sourceutν,ε+(F(uν,ε))x=νuxxν,ε,inR×(0,T), Turn MathJax on equation(1.8) uν,ε|t=0=gν,ε,uν,ε|t=0=gν,ε, Turn MathJax on where ν,εν,ε are positive constants and gν,εgν,ε is a regularized function of uIuI, εε is the scale of variation of shock positions. The solution to (1.7)–(1.8) can be viewed as an approximation of the solution to (1.1)–(1.2) as ν→0ν→0, and take the advantage of the quasi-shock configurations when they arise. Thus it is natural to study the sensitivity analysis by the method of vanishing viscosity. By the standard theory of parabolic equations (for instance, see [11] by Ladyzenskaya et al.), the solution uν,εuν,ε of (1.7)–(1.8) is smooth. Then the linearized one of (1.7) can be derived easily, which reads equation(1.9) View the MathML source(δuν,ε)t+(f(uν,ε)δuν,ε)x=ν(δuν,ε)xx,inR×(0,T), Turn MathJax on equation(1.10) δuν,ε|t=0=hν,ε,δuν,ε|t=0=hν,ε, Turn MathJax on where the initial datum hν,εhν,ε is a regularization of δuIδuI, the initial datum for (1.4). Here, gν,εgν,ε and hν,εhν,ε will be chosen suitably in Section 3, so that the perturbation of initial data and shock positions are taken into account. And we mollify and as follows equation(1.11) View the MathML source−ptν,ε−f(uν,ε)pxν,ε=νpxxν,ε,inR×(0,T), Turn MathJax on equation(1.12) View the MathML sourcepν,ε(x,T)=pν,εT(x),x∈R. Turn MathJax on Our main task is to investigate the limits of solutions to (1.7)– (1.8), (1.9)– (1.10) and (1.11)– (1.12), as ν,εν,ε tend to zero. Then natural questions arise as follows. 1. In order to take into account infinitesimal translation of shock positions and infinitesimal perturbation of initial data, and to recover the correct equation for the variation of shock positions and the limiting equations corresponding to (1.11), and respectively, can ν,εν,ε go to zero independently? And how about the convergent rates of solutions? 2. To solve the optimal control problem correctly, (1.4) should be complemented by the equation for δφδφ, the variation of shock positions. How about the existence of the latter equation? 3. Is it possible to apply efficient numerical methods, such as the “alternating descent method” developed in [1], to compute the minimizer for viscous optimal control problem in the quasi-shock case? To answer these questions, we shall employ the method of matched asymptotic expansions. We remark that this method could be generalized to the cases of multi-dimensional scalar equations and of 1-D systems. Our main results are the following. First , in order to construct asymptotic expansions which take into account infinitesimal translation of shock positions and infinitesimal perturbation of initial data, it is convenient to choose the parameters ν,εν,ε satisfying ε=σν,ε=σν, Turn MathJax on where σσ is a given positive constant. The equation (see Eq. (3.52)) of variation of shock positions differs from the one derived by Bressan and Marson (see [6]), etc., by a term which converges to zero as σσ tends to infinity. As a result, the perturbation of initial data plays a dominant role and the effect due to the artificial viscosity can be omitted. Second, we make use of matched asymptotic expansions to construct approximate solutions to problems (1.7)– (1.8) and (1.11)– (1.12), then prove that they converge, respectively, to the entropy solution and the reversible solution of the corresponding inviscid problems, while the solution to problem (1.9)– (1.10) converges to the one solving (1.4) in the sub-regions away from the shocks, and is complemented by an equation, i.e. (3.52), which governs the evolution of the variation of shock positions. The main convergence results are stated in Theorem 3.1 and Theorem 4.1. Third, by using the convergence results, we conclude that for numerical purpose, the alternating descent method is also applicable to optimal control problems in the cases of small viscosity (see Section 5). Fourth , we also clarify that some usual formal expansions in the theory of optimal control are only valid away from shocks. For example, one usually expands the solution uνuν to problem (1.7)– (1.8) as equation(1.13) uν=u+νδuν+O(ν2),uν=u+νδuν+O(ν2), Turn MathJax on where uu is the entropy solution to problem (1.1)– (1.2), and δuνδuν is the variation of uνuν, the solution to (1.9)– (1.10). However, the expansion (1.13) is not correct near the shocks. If δuν(x,0)δuν(x,0) is bounded, δuνδuν is continuous and uniformly bounded by the theory of parabolic equations. Since uνuν is also continuous, it follows from (1.13) that uu should be continuous too. It will lead to a contradiction if uu has shocks. Therefore, we should understand (1.13) as a multi-scale expansion, and assume that uν=uν(x,x/ν,t),δuν=δuν(x,x/ν,t)uν=uν(x,x/ν,t),δuν=δuν(x,x/ν,t). We shall obtain such an expansion by the method of matched asymptotic expansions. The new features to the method of asymptotic expansions in this article are mainly as follows. 1. Our expansions for uν,εuν,ε and δuν,εδuν,ε are different from the standard ones due to the fact that (1.7), (1.9) are not completely independent. In fact, when constructing asymptotic expansions we should find some compatible conditions for the asymptotic expansions of uν,εuν,ε and δuν,εδuν,ε, since (1.9) is the linearization of (1.7). 2. The approximate solution of pp should approach a constant in a “triangular region”, which is occupied by the characteristics forming the shock, due to the matching conditions (see (3.94), (3.95) etc.). 3. We derive the equation for the variation of shock positions from the outer expansions, but not from the inner expansions as the usual settings; see, e.g. [12]. Our approach is similar to the derivation of the Rankine–Hugoniot conditions. 4. In contrast to the paper by James and Sepúlveda (see [5]), we obtain a convergent rate for the adjoint problem; moreover, we only need the terminal data to be continuous , instead of W1,∞∩BVW1,∞∩BV. Notations: For any t>0t>0, we define Qt=R×(0,t)Qt=R×(0,t). C(t),Ca,…C(t),Ca,… denote, respectively, constants depending on t,a,…t,a,…, and CC is a universal constant. Let XX be a Banach space endowed with a norm ‖⋅‖X‖⋅‖X, and f:[0,T]→Xf:[0,T]→X. For any fixed tt the XX-norm of ff is denoted by ‖f(t)‖X‖f(t)‖X, especially, when X=L2(R)X=L2(R), we write ‖f(t)‖=‖f(t)‖X‖f(t)‖=‖f(t)‖X. Sometimes the variable tt is omitted for the sake of simplicity. Landau symbols O(1)O(1) and o(1)o(1). A quantity f(x,t;ν)=o(1)f(x,t;ν)=o(1) means ‖f‖L∞(Qt)→0‖f‖L∞(Qt)→0 as ν→0ν→0, and g(t;ξ)=o(1)g(t;ξ)=o(1) implies that ‖g‖L∞(0,t)→0‖g‖L∞(0,t)→0 as ξ→∞ξ→∞. And f(x,t;ν)=O(1)f(x,t;ν)=O(1) means ‖f‖L∞(Qt)≤C‖f‖L∞(Qt)≤C uniformly for ν∈(0,1]ν∈(0,1]. We also use the standard notations: BV(R)BV(R) (View the MathML sourceBVloc(R)), Lip(R)Lip(R) (View the MathML sourceLiploc(R)), are the sets of functions of (locally) bounded variations and the (locally) Lipschitz continuous functions in RR, respectively. The remaining part of this article is organized as follows. In Section 2, we collect some preliminaries and explain furthermore the motivation of this article. In Section 3, employing the method of matched asymptotic expansions and taking into account the infinitesimal perturbation of the initial datum and the infinitesimal translation of the shock positions, we shall construct, the inner and outer expansions, and obtain, by a suitable combination of these two expansions, the approximate solutions to problems (1.7)–(1.8), (1.9)–(1.10), and (1.11)–(1.12). Also the equations for the shock and its variation will be derived. In Section 4 we shall prove that the approximate solutions satisfy the corresponding equations asymptotically, and converge, respectively, to those of the inviscid problems in a suitable sense. Finally we discuss the alternating descent method in the context of viscous conservation laws in Section 5, where the convergence results will be used.