A Sensitivity and Adjoint Calculus for Discontinuous Solutions of Hyperbolic Conservation Laws with Source Terms

We present a sensitivity and adjoint calculus for the control of entropy solutions of scalar conservation laws with controlled initial data and source term. The sensitivity analysis is based on shift-variations which are the sum of a standard variation and suitable corrections by weighted indicator functions approximating the movement of the shock locations. Based on a first order approximation by shift-variations in L1 we introduce the concept of shift-differentiability which is applicable to operators having functions with moving discontinuities as images and implies differentiability for a large class of tracking-type functionals. In the main part of the paper we show that entropy solutions are generically shift-differentiable at almost all times t > 0 with respect to the control. Hereby we admit shift-variations for the initial data which allows to use the shift-differentiability result repeatedly over time slabs. This is useful for the design of optimization methods with time domain decomposition. Our analysis, especially of the shock sensitivity, combines structural results by using generalized characteristics and an adjoint argument. Our adjoint based shock sensitivity analysis does not require to restrict the richness of the shock structure a priori and admits shock generation points. The analysis is based on stability results for the adjoint transport equation with discontinuous coefficients satisfying a one-sided Lipschitz condition. As a further main result we derive and justify an adjoint representation for the derivative of a large class of tracking-type functionals. Key words. optimal control, scalar conservation law, shock sensitivity, adjoint state, Fréchet differentiability AMS subject classifications. 35L65, 49J50, 49K20, 35R05, 35B37 1. Introduction. This paper is concerned with the development of a sensitivity and adjoint calculus for the optimal control of entropy solutions of scalar conservation laws with source terms: Consider the Cauchy problem for an inhomogeneous conservation law ty + xf(y) = g(t; x; y; u1); (t; x) 2 T def = ℄0; T [ R y(0; x) = u0(x); x 2 R; (1.1) where the flux f : R ! R is twice continuously differentiable, u = (u0; u1) 2 L1(R) L1( T )m def = U is the control and g : T R Rm ! R satisfies a growth condition (A1) g 2 L1( T ;C0;1 lo (R Rm)) and for all Mu > 0 there are C1; C2 > 0 with g(t; x; y; u1) sgn(y) C1 + C2 jyj; 8 (t; x; y; u1) 2 T R [ Mu;Mu℄m: Then one can show (e.g., [18, 25]) that (1.1) admits for all u 2 U a unique entropy solution y = y(u) 2 L1( T ) \ C([0; T ℄;L1lo (R)). It is well known that in general weak solutions of (1.1) develop discontinuities after finite time and that uniqueness holds only in the class of entropy solutions. We recall that for given u 2 U a function y = y(u) 2 L1( T ) is an entropy solution to (1.1) in the sense of Kružkov [18] if it satisfies the entropy inequality t (y) + xq(y) 0(y)g(t; x; y; u1) in D0( T ) for all convex functions (entropies) : R ! R with corresponding entropy fluxes q(y) = R y 0 0(s) f 0(s) ds and the initial condition in the sense ess lim t!0+ ky(t; ) u0k1;K d = 0 8K R: The aim of this paper is to develop and justify – without a priori assumptions on the shock structure – a sensitivity calculus for the control-to-state mapping u 7 ! y(u) that yields in particular the differentiability and a formula for the derivative of objective functionals (1.2) J(y(u)) = Z b a (y( t; ;u); yd) dx Zentrum Mathematik, Technische Universität München, 80290 München, Germany ([email protected]). The author was supported by Deutsche Forschungsgemeinschaft under Grant Ul158/2-1 and by CRPC grant CCR– 9120008. 1 A SENSITIVITY AND ADJOINT CALCULUS FOR CONSERVATION LAWS 2 with data yd 2 BV ([a; b℄), 2 C1;1 lo (R2 ) and t 2℄0; T ℄. Moreover, we will derive an adjoint formula for the gradient of (1.2). The adjoint equation is a transport equation with source term and its coefficient is discontinuous along shock curves which requires a careful definition of the adjoint state as a reversible solution to ensure uniqueness and stability. These results are useful for the design of gradient based methods for the solution of control problems of the type (P) min ~ J(y(u); u) def = J(y(u)) + R(u) subject to u 2 Uad; y(u) solves (1.1): In [25] we have derived results on the existence of optimal solutions for (P) and the convergence of discretized approximations for the multidimensional case. For example, (P) has an optimal solution if Uad is bounded in L1(R) L1( T )m and compact in L1lo (R) L1lo ( T )m and if ~ J : C([0; T ℄;Lrlo (R)) (Uad Lrlo (R) Lrlo ( T )m) ! R is sequentially lower semicontinuous for some r 2 [1;1[. Moreover, for the present one dimensional case existence results without compactness assumption on Uad were obtained in [25] using compensated compactness. The state equation (1.1) is a useful model for the study of control problems involving flows with shocks. In particular, it is shown in [10] that the steady flow of an inviscid fluid in a duct governed by the Euler equations can be reduced to determining the velocity y as a steady state solution of (1.1) for f(y) = y + 2 H=y, g(x; y; u1) = u1 (y 2H=y), where H is the total enthalpy, = ( 1)=( + 1) with the gas constant > 1 and the design variable u1 = xA=A, A(x) being the cross-sectional area of the duct. Moreover, it is noted in [10] that the corresponding time-dependent problem (1.1) captures some essential features of the time-dependent Euler equations and is therefore a suitable model problem for the study of unsteady duct flows with shocks. Since the flow over a transonic airfoil is qualitatively similar to one-dimensional duct flows, the study of the differentiability properties of (1.1)–(1.2) is thus useful to gain insight into the optimal design of airfoils under unsteady flow conditions. In particular, the sensitivity of flows with shocks with respect to time-dependent changes of the geometry is of practical importance for the control of systems with fluid-structure coupling, e.g., the fluttering problem of airfoils. In this work we give a rigorous sensitivity analysis and adjoint calculus for solutions of (1.1) with shocks and thereby provide an analytical framework for the study and numerical solution of optimal control problems governed by hyperbolic balance laws (1.1). The main features of our approach are: we derive a sensitivity calculus based on shift-variations that implies the Fréchet differentiability for objective functionals (1.2), we give and rigorously justify a gradient representation for objective functionals (1.2) by using an appropriately defined adjoint state, the shock structure has not to be restricted a priori and shock generation points are allowed, we admit non-entropy-admissible initial data and allow shift-variations of the initial data that move shock locations. The crucial part is the analysis of the shock sensitivities. In our approach we derive the differentiability of a shock position xs at time t by analyzing the smoothness properties of the functional u 7 ! R xs+" xs " y( t; x;u) dx with the help of an adjoint calculus for " ! 0. The adjoint argument is mainly based on the stability of the adjoint equation with respect to its coefficients. Then the properties of u 7 ! xs(u) follow from basic stability properties of the shock that we derive a priori by using the theory of generalized characteristics. An advantage of this method lies in the fact that the shock structure of the solution has not to be restricted a priori. In particular, shock generation points are allowed. This approach can, at least in a formal manner, also be applied to hyperbolic systems and gives the correct shock sensitivity if the necessary stability properties of the shock and the adjoint equation actually hold. It can be shown (see, e.g., [25] for the problem at hand) that the mappingu 2 U 7 ! y(u) 2 C([0; T ℄;L1lo (R)) is locally Lipschitz, but very simple examples show that this mapping is in general not directionally differentiable if y(u) contains shocks, even if L1 is replaced by C1 A SENSITIVITY AND ADJOINT CALCULUS FOR CONSERVATION LAWS 3 in the definition of U . This is caused by the fact that a variation of u results in a shift of the shocks, which can not be approximated appropriately in the linear structure of L1lo . Consider for example the following family of Riemann problems for the inviscid Burgers equation ty" + x y2 " 2 = 0; y"(0; x) = u0(x) + "Æu0(x) def = (1 + " if x 0 1 if x > 0 The solution has a shock " emanating from (0; 0) with shock speed _ "(t) = [(y"(t; "(t)))2=2℄ [y"(t; "(t))℄ = " 2 ; according to the jump condition, where [h(t; x)℄ = h(t; x ) h(t; x+) denotes the jump of h(t; ) across x. Thus, we have "(t) = " 2 t and the corresponding entropy solution y" = (1 + " if x t"=2 1 if x > t"=2 Of course, the Lipschitz continuous map " 7 ! y"(t; ) 2 L1lo (R) is not differentiable, since the difference quotient does only converge in a weaker topology, for example weakly in the space Mlo (R) of locally bounded Borel measures. In fact, " 7 ! y"(t; ) 2 Mlo (R) is differentiable in the weak topology and we have for example at " = 0 (1.3) d d"y"(t; )j"=0 " = 1fx< 0(t)g "+ [y0(t; 0(t))℄Æ 0(t) t 2 " where 1I denotes the indicator function of a set I , i.e., 1I(x) = 1 if x 2 I , 1I(x) = 0 else, and Æx denotes the Dirac measure located at x. Hereby, t 2 " is a linear (in this case exact) approximation of the actual shock shift "(t) 0(t). Note however, that a differentiability result in the weak topology of Mlo (R) is not strong enough to derive the differentiability of the functional (1.2) without additional structural information. To get a first order approximation in L1lo , we have to leave the linear structure of L1lo in order to allow for an accurate approximation of the shock movement. A natural way to achieve this is to replace the singular (second) part of the measure (1.3) by the function sgn( t 2")[y0(t; 0(t))℄1I( 0(t); 0(t)+ t 2 "), where I(a; b) def = [minfa; bg;maxfa; bg℄ and sgn( ) is the sign function. We thereby obtain a first order approximation of y"(t) y0(t) in L1lo by the shift variation S 0(t) y0(t)(1fx< 0(t)g"; t 2") def = 1fx< 0(t)g "+ sgn( t 2")[y0(t; 0(t))℄1I( 0(t); 0(t)+ t 2 "): In this paper we will develop a sensitivity calculus based on shift-variations for the mapping u 7 ! y( t; ;u), t 2 [0; T ℄, defined in (1.1) in the case f 00 mf 00 > 0. Let piecewise C1 initial data u0 2 PC1(R; x1 ; : : : ;xN ) and u1 2 L1(0; T ;C1(R)m) be given. Using the theory of generalized characteristics [8] we will show that for a given interval I and time t > 0 the following situation is generic: y( t; ;u) has on I finitely many shocks at x1 < : : : < xK , the shock locations depend differentiable on u and the states connected by the shocks vary differentiable in the strong topology of L1. From this we will deduce that the variation y( t; ;u + Æu) y( t; ;u) allows a first order approximation by a shift variation of the form (1.4) S( xk) y( t; ;u)(Æy; s) def = Æy +Xk [y( t; xk;u)℄ sgn( sk)1I( xk; xk+ sk) where (Æy; s) depends linearly on Æu, xk are the shock locations, [y( t; xk;u)℄ = y( t; xk ;u) y( t; xk+;u) denotes the jump across the shock, I( xk; xk + sk) is the interval enclosed by the arguments, 1I( xk; xk+ sk) is its indicator function, and sk is a linear approximation of the shock A SENSITIVITY AND ADJOINT CALCULUS FOR CONSERVATION LAWS 4 shift. To mimic the behavior of y( ;u) at later times, it is natural to go one step further and to admit shift-variations already for the initial data. Roughly speaking, we will in particular show that for (u0; u1) as above,W def = PC1(R;x1 ; : : : ; xN ) L1(0; T ;C1(R)m) RN and almost all t the mapping (1.5) (w0; w1; ) 2W 7 ! y( t; ;u0 + S(xi) u0 (w0; ); u1 + w1) 2 L1(I) is shift differentiable with respect to (w0; w1; ) at 0 in the sense that its variation admits a first order approximation by a shift variation of the form (1.4), where (Æy; s) depends linearly on Æw0; Æw1, and s = Æ . Hereby, Æy can be obtained as the trace ÆY ( t) of a function ÆY that is the piecewise solution of the linearization of (1.1) outside of the shock set and sk can be obtained by an adjoint formula. We admit shift variations of the initial data since this allows to use the shift-differentiability result repeatedly over time slabs. This is helpful for the design of optimization algorithms with time domain decomposition for the solution of (P). By introducing a general concept of shift-differentiability we will be able to derive results on the Fréchet differentiability of tracking-type functionals of the form (1.2) as long as the discontinuities of yd and y( t; ;u) do not coincide. If yd and y( t; ;u) share discontinuities we will still obtain directional differentiability. For objective functionals of the form (1.2) we will derive a gradient representation via an adjoint state. The proper definition of the adjoint state requires an extension of the concept of reversible solutions of backward transport equations with discontinuous coefficients introduced in [1] to the case tp+ f 0(y) xp = gy(t; x; y; u1) p; (t; x) 2 t def = ℄0; t[ R p( t; x) = p t(x); x 2 R (1.6) with linear source term, where p t are suitable end data. The results of this paper can be straightforward extended to identification problems for the flux f , where f is the control. Identification problems of this type are considered by James and Sepúlveda [17]. The differentiability of the objective function for the hyperbolic case was left open. The techniques of the present paper can be used to obtain a sensitivity and adjoint calculus for flux identification as well. In recent years several results on sensitivities and adjoints for hyperbolic conservation laws were obtained by other authors [1–6, 12, 13] but most results assume a priori knowledge of the shock structure (usually one shock separating smooth states) or are restricted to the conservative case g 0. The conservative case admits special techniques, since the characteristics are straight lines and the solution can be represented by the integral formula of Lax [19]. Bouchut and James apply in [2] their existence and stability results of [1] for measure-valued duality solutions of linear conservation laws with discontinuous coefficients to derive for the case g 0; f 00 > 0 that u0 2 L1 7 ! y( ;u) 2 C([0; T ℄;Mlo (R)–w ) is directionally differentiable at an entropy-admissible u0 where the spaceMlo (R) of local Borel measures is equipped with the usual weak topology. Note that this topology is too weak to derive directly differentiability results for (1.2) without using additional structural information. Godlewski and Raviart study in [12] (see also [13]) the linearized stability of multidimensional hyperbolic systems of conservation laws for perturbations of the initial data of a base solution with a one dimensional shock. They define measure solutions for the linearized equations with singular part along the shock and construct numerical schemes for the solution of the linearized problem. For the conservative scalar problem with Riemann initial data it is shown that the linearization coincides with the first order expansion in C([0; T ℄;Mlo (R)–w ) of [2]. In this paper we give further justification of this linearization process for more general situations. Bouchut and James develop in [1] existence and stability results for transport equations with discontinuous coefficients satisfying a one sided Lipschitz condition that will be extended in the present work for the analysis of the adjoint equation (1.6). Previous results on the adjoint equation were obtained in the context of uniqueness results in [7, 16, 20, 21], and of error estimates for A SENSITIVITY AND ADJOINT CALCULUS FOR CONSERVATION LAWS 5 approximate solutions in [24]. In [20] adjoint equations for a class of systems of conservation laws are considered. An extension of our approach to systems seems to be possible by building on this work. In [3] a new differential structure on the space BV obtained by horizontal shifts of the points of the graph is introduced and it is shown that in the case g 0; f 00 > 0 the flow u0 2 L1 7 ! y(t; ;u) generated by (1.1) is generically differentiable w.r.t. this structure. The analysis uses the integral formula of Lax. Bressan and Marson [4] use generalized tangent vectors to develop a variational calculus for piecewise Lipschitz solutions of systems of conservation laws. Using our notation (1.4), they show that for piecewise Lipschitz initial data u"0, " 0, such that u"0 u0 = S(xi) u0 ("Æu0; "s) + o(") in L1lo , the corresponding solutions y" satisfy y"( t; ) y( t; ) = S( xk) y( t; )("Æy; " s) + o(") in L1lo if t is so small that y" remains piecewise Lipschitz on [0; t℄. While this result applies to systems it considers only directional variations and requires the structural assumption of piecewise Lipschitz solutions which is not needed in the present paper. Especially the analysis of the shock sensitivity differs significantly from our approach, since in [4] the linearized Rankine-Hugoniot jump condition together with the linearized state equation is used to derive an ODE for the shock sensitivity, while we use an adjoint formula which reduces the necessary structural information on the history of the shock as far as possible. Moreover, we develop an adjoint calculus that gives a gradient representation for objective functionals (1.2). Cliff, Heinkenschloss, and Shenoy study in [5, 6] design problems for one-dimensional steady duct flow. By introducing the single shock location as additional state variable and transforming the space variable such that the shock location is fixed Fréchet differentiability is shown. Optimality conditions are derived and an adjoint-based gradient representation of the objective function is given. Finally, numerical results for the application of an SQP method to the discretized problem are reported. This paper is organized as follows. In section 2 we introduce the concept of shift-differentiability for operators having discontinuous functions with moving discontinuities as images which is based on a first order approximation by shift-variations (1.4). Moreover, we will show that the superposition (1.2) of a shift-differentiable operator u 7 ! y( t; ;u) and a tracking-type functional is Fréchet differentiable if yd; y( t; ;u) do not share discontinuities and is directionally differentiable else. In section 3 we state the main results of the paper. In 3.1 we state in Theorem 3.2 a shift-differentiability result for entropy solutions of (1.1) w.r.t the controls, more precisely of the control-state-mapping (1.5), if a nondegeneracy assumption holds for all shocks at the observation time t. In Theorem 3.4 we give a formula for the corresponding shift-derivative. Moreover, we sketch the main line of the proofs. The next Theorem 3.8 shows that the required nondegeneracy assumption holds for all shocks at almost all times t > 0 if u0 2 PC2 andu1 2 L1(0; T ;C2 lo ). In 3.2 the results of 3.1 and the general shift-differentiability calculus are combined to obtain in Theorem 3.9 and Corollary 3.10 the differentiability of tracking-type functionals u 7! J(u) in (1.2). In 3.3 we finally state a convenient adjoint-based gradient representation for the derivative of these objective functionals with respect to the inner product of L2, see Theorem 3.11. The proofs of these main results are prepared in sections 4–8 and finally carried out in section 9. Section 4 provides the necessary stability results and collects structural results of [8] provided by the theory of generalized characteristics. We use this to derive basic differentiability results for the solution along generalized characteristics. In section 5 continuity points are analyzed that are no shock generation points. In 5.1 and 5.2 we study continuity points in the exterior and interior of rarefaction waves. In 5.3 continuity points are analyzed that are located on characteristics emanating from points where discontinuities are produced under shift-variations and 5.4 studies points on the boundary of rarefaction waves. In section 6 the stability of shocks and the differentiability of the shock location at a time t > 0 is shown under a nondegeneracy assumption. The proof of the latter is carried out in section 8, since it requires stability results for the adjoint equation which are provided in section 7. In section 9 we prove the main results already stated in section 3 by combining the results of the previous sections. Conclusions and future work are presented in 10. The Appendix contains a proof of the results in section 7 on the adjoint equation. A SENSITIVITY AND ADJOINT CALCULUS FOR CONSERVATION LAWS 6 Notations. For Lebesgue-measurable S Rn the norm of the Lebesgue-spaces Lr(S), 1 r 1, is denoted by k kr;S . In the case S = Rn we write k kr. By ( ; )2;S we denote the inner product on L2(S). For an interval I R the space of functions v 2 L1(I) with bounded variation jvjvar is denoted by BV (I). For open S Rn we mean by Ck(S), k 2 N0 , the space of functions with continuous, bounded derivatives on S up to order k equipped with the usual norm kvkCk(S) = Pj j k kD vk1;S . Ck(S l) is the subspace of functions in Ck(S) that admit a continuous extension of the first k derivatives to S l. Moreover, we write C(S) instead of C0(S). Ck; (S l), 0 < 1, is the usual Hölder space. For closed I [a; b℄, a < b, we denote by PCk(I;x1; : : : ; xN ) the space of piecewise Ck-functions v with possible discontinuities at a < x1 < : : : < xN < b, more precisely vjIi 2 Ck(I l i ), i = 0; : : : ; N , with Ii =℄xi; xi+1[, i = 1; : : : ; N 1, I0 = I\fx < x1g, IN = I\fx > xNg. It is endowed with the norm kvkPCk(I;x1;:::;xn) =PNi=0 kvkCk(I l i ). The indicator function of a set I is denoted by 1I , i.e., 1I(x) = 1 if x 2 I and 1I(x) = 0 else. 2. Shift-differentiability. In this section we introduce a concept of shift-differentiability that yields an extension of classical differentiability to operators u 7 ! y(u) 2 L1lo (R) having functions with moving discontinuities as images. It is based on shift-variations that are the sum of a standard variation and suitably scaled indicator functions approximating the actual shift of discontinuities. The interesting point is that the shift-differentiability of an operator implies under quite general circumstances the Fréchet differentiability of tracking-type functionals analogously to (1.2). 2.1. Shift-variations and shift-differentiability of operators. As motivated in the introduction we define shift-variations as follows. Definition 2.1. Let I = [a; b℄, a < b and let w 2 BV (I). Given a < x1 < x2 < : : : < xN < b, we call (Æw; s) 2 L1(I) RN a generalized variation of w and associate with it the shift-variation S(xi) w (Æw; s) 2 L1(R) of w by S(xi) w (Æw; s)(x) def = Æw(x) + N Xi=1[w(xi)℄+ sgn(si)1I(xi;xi+si)(x) where [w(xi)℄+ def = max f0; w(xi ) w(xi+)g and I( ; ) def = [minf ; g;maxf ; g℄ for ; 2 R. The restriction that only down-jumps are shifted is motivated by the fact that entropy solutions of (1.1) satisfy Oleinik’s entropy condition y(t; x ) y(t; x+) for all x 2 R and a.a. t 2℄0; T ℄, see section 4.1 below. We call an operator shift-differentiable, if its actual variation admits a first order approximation in L1lo by a shift-variation, more precisely Definition 2.2. Let U be a real Banach space and I = [a; b℄, a < b. For an open subset D U let u 2 D 7 ! y(u) 2 L1(R) be locally bounded. Moreover, let u 2 D with y( u) 2 BV (I). We say that y is shift-differentiable at u if there are a < x1 < x2 < : : : < xK < b and a bounded linear operator Ts( u) = Dsy( u) 2 L(U;Lr(I) RK ), r 2℄1;1℄, such that lim u! u ky(u) y( u) S( xk) y( u)(Ts( u) (u u))k1;I ku ukU = 0: We say that y is continuously shift-differentiable at u if y is shift-differentiable in a neighborhood of u and if the corresponding Ts(u), xk(u), k = 1; : : : ;K , as well as y(u)( xk(u) ) are continuous at u. 2.2. Differentiability after composition with cost functionals. The property of shiftdifferentiability is strong enough that it implies the Fréchet differentiability of functionals of A SENSITIVITY AND ADJOINT CALCULUS FOR CONSERVATION LAWS 7