Stability of nonlinear locally damped partial differential equations: the continuous and discretized problems. Part I

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1 . Stability of nonlinear locally damped partial differential equations: the continuous and discretized problems. Part I Fatiha Alabau-Boussouira 1 Emmanuel Trélat 2 1 Univ. de Lorraine, LMAM 2 Univ. Paris 6 (LJLL) and IUF 33th International Summer School of Automatic Control, Grenoble, september

2 Outline of the first lecture 1 General motivations for dissipative vibrating systems 2 Dissipative systems in finite dimensions 3 Infinite dimensional dissipative systems 4 The scalar case 5 The optimal-weight convexity method 6 Lower energy estimates: an energy comparison principle 7 Examples of decay rates and optimality 8 Vectorial case: parameter-dependent upper estimates

3 General motivations for dissipative vibrating systems Sommaire 1 General motivations for dissipative vibrating systems 2 Dissipative systems in finite dimensions 3 Infinite dimensional dissipative systems 4 The scalar case 5 The optimal-weight convexity method 6 Lower energy estimates: an energy comparison principle 7 Examples of decay rates and optimality 8 Vectorial case: parameter-dependent upper estimates

4 General motivations for dissipative vibrating systems Many physical phenomenon s, such as the propagation of waves in a string, a membrane or a plate are modelized by finite or infinite dimensional vibrating systems. In applications, the engineers want to reduce to zero the vibrations of the solutions by auto-regulation. This is performed through the implementation of appropriate feedbacks on selected regions of the physical device. The auto-regulated system is a damped system and the vibrations of its solutions are in general measured through their energies.

5 General motivations for dissipative vibrating systems We consider reversible dissipative systems in both finite and infinite dimensions in autonomous or non autonomous form, meaning that the damping effect can be nonlocal. The common feature for dissipative systems is : one can associate a (physical) energy to them moreover this energy is decaying as time increases. Damping (feedback) = the energy is decaying

6 General motivations for dissipative vibrating systems We consider reversible dissipative systems in both finite and infinite dimensions in autonomous or non autonomous form, meaning that the damping effect can be nonlocal. The common feature for dissipative systems is : one can associate a (physical) energy to them moreover this energy is decaying as time increases. Damping (feedback) = the energy is decaying

7 General motivations for dissipative vibrating systems We consider reversible dissipative systems in both finite and infinite dimensions in autonomous or non autonomous form, meaning that the damping effect can be nonlocal. The common feature for dissipative systems is : one can associate a (physical) energy to them moreover this energy is decaying as time increases. Damping (feedback) = the energy is decaying

8 General motivations for dissipative vibrating systems We consider reversible dissipative systems in both finite and infinite dimensions in autonomous or non autonomous form, meaning that the damping effect can be nonlocal. The common feature for dissipative systems is : one can associate a (physical) energy to them moreover this energy is decaying as time increases. Damping (feedback) = the energy is decaying

9 General motivations for dissipative vibrating systems We consider reversible dissipative systems in both finite and infinite dimensions in autonomous or non autonomous form, meaning that the damping effect can be nonlocal. The common feature for dissipative systems is : one can associate a (physical) energy to them moreover this energy is decaying as time increases. Damping (feedback) = the energy is decaying

10 General motivations for dissipative vibrating systems The energy measures the vibrations of the device, thus it is important to determine its asymptotic behavior at large time. Questions: Is the damping sufficient to induce strong stabilization (the energy goes to 0 at )? This is known to hold in general. References: Dafermos, Ball, Slemrod... in the 70 s, beggining of 80 s What about decay rates and how they depend on the feedback properties? Unified approach between feedbacks types? Optimality of the decay rates? Links/differences between the finite and infinite dimensions? Discretization of systems of infinite dimensional vibrating systems and links between the continuous and discretized systems?

11 General motivations for dissipative vibrating systems The energy measures the vibrations of the device, thus it is important to determine its asymptotic behavior at large time. Questions: Is the damping sufficient to induce strong stabilization (the energy goes to 0 at )? This is known to hold in general. References: Dafermos, Ball, Slemrod... in the 70 s, beggining of 80 s What about decay rates and how they depend on the feedback properties? Unified approach between feedbacks types? Optimality of the decay rates? Links/differences between the finite and infinite dimensions? Discretization of systems of infinite dimensional vibrating systems and links between the continuous and discretized systems?

12 Dissipative systems in finite dimensions Sommaire 1 General motivations for dissipative vibrating systems 2 Dissipative systems in finite dimensions 3 Infinite dimensional dissipative systems 4 The scalar case 5 The optimal-weight convexity method 6 Lower energy estimates: an energy comparison principle 7 Examples of decay rates and optimality 8 Vectorial case: parameter-dependent upper estimates

13 Dissipative systems in finite dimensions The scalar case : where Dissipative systems in finite dimensions u + ν u + f (u) + ρ(u ) = 0. ν > 0 u is a scalar unknown f is locally lipschitz continuous and sf (s) 0 for all s R ρ is monotone increasing, ρ(0) = 0. Multiply the equation by u. This gives the so-called dissipation relation: where ( 1 2( u (t) 2 + ν u(t) 2) + F(u(t))) = u (t)ρ(u (t))., F (u) = u 0 f (s) ds.

14 Dissipative systems in finite dimensions The scalar case : where Dissipative systems in finite dimensions u + ν u + f (u) + ρ(u ) = 0. ν > 0 u is a scalar unknown f is locally lipschitz continuous and sf (s) 0 for all s R ρ is monotone increasing, ρ(0) = 0. Multiply the equation by u. This gives the so-called dissipation relation: where ( 1 2( u (t) 2 + ν u(t) 2) + F(u(t))) = u (t)ρ(u (t))., F (u) = u 0 f (s) ds.

15 Dissipative systems in finite dimensions Define the energy E(t) = 1 2 ( u (t) 2 + ν u(t) 2) + F(u(t)), Kinetic energy Then the dissipation relation becomes potential energy E (t) = u (t)ρ(u (t)). Since sρ(s) 0 for all s R, we have E (t) = u (t)ρ(u (t)) 0 t 0. = the energy is a Lyapunov function, however it is not a strict Lyapunov function, since the dissipation relation does not involve the potential part of the energy.

16 Dissipative systems in finite dimensions The vectorial case: where the unknown u R n and u + Au + f (u) + Bρ(u ) = 0. A is a symmetric, positive definite matrix B = diag(b i ) 1 i n, with b i 0 i {1,..., n} (f (u)) i = f (u i ), (ρ(u )) i = ρ(u i ) Take the euclidian scalar product of the above equality with u, and denote by the euclidian norm. This gives n n u, u + Au, u + (F(u i )) = b i ρ(u i )u i. where F = f as for the scalar case. i=1 i=1

17 Dissipative systems in finite dimensions The vectorial case: where the unknown u R n and u + Au + f (u) + Bρ(u ) = 0. A is a symmetric, positive definite matrix B = diag(b i ) 1 i n, with b i 0 i {1,..., n} (f (u)) i = f (u i ), (ρ(u )) i = ρ(u i ) Take the euclidian scalar product of the above equality with u, and denote by the euclidian norm. This gives n n u, u + Au, u + (F(u i )) = b i ρ(u i )u i. where F = f as for the scalar case. i=1 i=1

18 Dissipative systems in finite dimensions Thus, for this example the energy is given by E(t) = 1 2 ( u 2 + Au, u + n ) (F(u i )), and we have the dissipation relation (under the same hypothesis on ρ than in the scalar case) i=1 n E (t) = b i ρ(u i )u i 0 t 0. i=1 Thus, the energy is again a Lyapunov function, but not a strict one.

19 Dissipative systems in finite dimensions The motivation for vectorial system comes from the semi-discretization of wave-type equations. Let us consider a frictional dissipative wave equation in the one-dimensional space domain (0, 1). t 2u 2 x u + f (u) + b(x)ρ(x, t u) = 0, 0 < t, 0 < x < 1, u(t, x) = 0, for x = 0, x = 1, 0 < t, u(0, x) = u 0 (x), t u(0, x) = u 1 (x), 0 < x < 1. (1) We assume that this system is dissipative, i.e. b 0 is the damping coefficient (locally supported in general), ρ is monotone nondecreasing with respect to the second variable and ρ(., 0) = 0. Additional hypotheses on f to guarantee existence for all t 0.

20 Dissipative systems in finite dimensions A semi-discretization of the above equation in space, with for instance a uniform mesh x i = i h for i = 0,..., n + 1 with a parameter of discretization h = 1/(n + 1) gives the finite dimensional system u i u i+1 + u i 1 2u i h 2 + f (u i ) + b i ρ i (u i ) = 0, 0 < t, i = 1,..., n, u 0 (t) = u n+1 (t) = 0, 0 < t, u i (0) = u i,0, t u i (0) = u i,1, i = 1,..., n, (2) where u i is a function of t which stands for an approximation of the solution u at point x i b i = b(x i ), ρ i (s) = ρ(x i, s) for all s R.

21 Dissipative systems in finite dimensions A semi-discretization of the above equation in space, with for instance a uniform mesh x i = i h for i = 0,..., n + 1 with a parameter of discretization h = 1/(n + 1) gives the finite dimensional system u i u i+1 + u i 1 2u i h 2 + f (u i ) + b i ρ i (u i ) = 0, 0 < t, i = 1,..., n, u 0 (t) = u n+1 (t) = 0, 0 < t, u i (0) = u i,0, t u i (0) = u i,1, i = 1,..., n, (2) where u i is a function of t which stands for an approximation of the solution u at point x i b i = b(x i ), ρ i (s) = ρ(x i, s) for all s R.

22 Dissipative systems in finite dimensions A semi-discretization of the above equation in space, with for instance a uniform mesh x i = i h for i = 0,..., n + 1 with a parameter of discretization h = 1/(n + 1) gives the finite dimensional system u i u i+1 + u i 1 2u i h 2 + f (u i ) + b i ρ i (u i ) = 0, 0 < t, i = 1,..., n, u 0 (t) = u n+1 (t) = 0, 0 < t, u i (0) = u i,0, t u i (0) = u i,1, i = 1,..., n, (2) where u i is a function of t which stands for an approximation of the solution u at point x i b i = b(x i ), ρ i (s) = ρ(x i, s) for all s R.

23 Dissipative systems in finite dimensions Set A = h which is symmetric and positive definite. We can rewrite the semi-discretized equation as the vectorial system where the unknown u R n u + Au + f (u) + ρ(u ) = 0. In a similar way, we can consider a semi-discretization of the plate (or Petrowsky) equation.

24 Dissipative systems in finite dimensions Thus, semi-discretized infinite dimensional vibrating systems enter in the framework of finite dimensional vibrating systems. We want optimal energy decay rates for the scalar and vectorial systems. As a consequence, this would also give optimal decay rates for semi-discretized systems. But in addition, it is important for applications to trace the dependence of the estimates on the discretization parameters and to obtain numerical schemes which lead to uniform decay rates with respect to h.

25 Dissipative systems in finite dimensions Thus, semi-discretized infinite dimensional vibrating systems enter in the framework of finite dimensional vibrating systems. We want optimal energy decay rates for the scalar and vectorial systems. As a consequence, this would also give optimal decay rates for semi-discretized systems. But in addition, it is important for applications to trace the dependence of the estimates on the discretization parameters and to obtain numerical schemes which lead to uniform decay rates with respect to h.

26 Dissipative systems in finite dimensions The discrete energy is defined by E h (t) = h 2 n { (u i ) 2 (t) + ( u i+1 u i ) } 2 h i=0 and satisfies, in the linear damped, i.e. when ρ(., s) = a(.)s, the discrete dissipation relation E h(t) = h n a i (u i ) 2 (t) i=1

27 Dissipative systems in finite dimensions It is well-known that in the linear damped case, the above numerical scheme leads to nonuniform exponential decay rates with respect to h. These results are strongly related to observability estimates for the undamped equation, which are not uniform with respect to h. These results are due to Glowinski-Li-Lions, Glowinski, Infante-Zuazua, Tebou-Zuazua... This phenomenon is due to high frequency numerical spurious oscillations. Thus in the nonlinear damped case, the first step is to derive optimal energy decay rates which are h-dependent then check how to modify the numerical scheme to derive optimal energy decay rates that are uniform with respect to h.

28 Infinite dimensional dissipative systems Sommaire 1 General motivations for dissipative vibrating systems 2 Dissipative systems in finite dimensions 3 Infinite dimensional dissipative systems 4 The scalar case 5 The optimal-weight convexity method 6 Lower energy estimates: an energy comparison principle 7 Examples of decay rates and optimality 8 Vectorial case: parameter-dependent upper estimates

29 Infinite dimensional dissipative systems Infinite dimensional systems We consider the following abstract equation u + Au + feedback operator[u] = 0 (u, u )(0) = (u 0, u 1 ) Here A stands for an unbounded linear operator in an Hilbert space H, which is closed, coercive self-adjoint with dense domain in H.

30 Infinite dimensional dissipative systems As for the finite dimensional case, one can associate an energy to the solutions and this energy decays through time. The same questions than for the finite dimensional case arise, except now that there is a space dependence and the problem is set-up in an infinite dimensional framework (Sobolev spaces) Moreover the feedback can be locally or boundary supported and in general the support of the feedback coefficient has to satisfy geometric conditions. We shall come back later on infinite dimensional systems.

31 Infinite dimensional dissipative systems As for the finite dimensional case, one can associate an energy to the solutions and this energy decays through time. The same questions than for the finite dimensional case arise, except now that there is a space dependence and the problem is set-up in an infinite dimensional framework (Sobolev spaces) Moreover the feedback can be locally or boundary supported and in general the support of the feedback coefficient has to satisfy geometric conditions. We shall come back later on infinite dimensional systems.

32 The scalar case Sommaire 1 General motivations for dissipative vibrating systems 2 Dissipative systems in finite dimensions 3 Infinite dimensional dissipative systems 4 The scalar case 5 The optimal-weight convexity method 6 Lower energy estimates: an energy comparison principle 7 Examples of decay rates and optimality 8 Vectorial case: parameter-dependent upper estimates

33 The scalar case Let us start by the scalar case example: u + ν u + f (u) + ρ(u ) = 0. where ν > 0 u is a scalar unknown f is locally lipschitz continuous, sf (s) 0 for all s R µ > 0 such that 0 F (s) µsf (s), s R, with F(u) = u ρ is monotone increasing, ρ(0) = 0 0 f (s) ds

34 The scalar case We assume that the feedback satisfies the assumption ρ C(R), is monotone increasing ρ(0) = 0 a strictly increasing odd function g such that (A1) c s ρ(s) C s, s 1 c g( s ) ρ(s) C g 1 ( s ), s 1 r 0 > 0 such that g C 1 ([0, r 0 ]), g(0) = g (0) = 0 where g 1 denotes the inverse function of g and where c, C are positive constants. Remark The function g has an arbitrary sublinear growth the interesting case close to 0. It captures the behavior of the feedback close to 0.

35 The scalar case Remark If g (0) 0 then it is as for the linear feedback case. So it will not be considered here. The assumption of linear growth at infinity is made for the sake of simplicity. It can be removed using a well-known strong stabilization result (based on Lasalle invariance principle) as follows E(t) 0 as t. Thus for sufficiently large time u (t) r 0.

36 The scalar case The energy is defined as E(t) = 1 2 ( u (t) 2 + ν u(t) 2) + F(u(t)), where F (u) = u 0 f (s) ds. Recall the dissipation relation: E (t) = u (t)ρ(u (t)). where E is the energy of the solution.

37 The scalar case We suppose f 0, ν = 1. Then E is given by E(t) = 1 2 ( u (t) 2 + ν u(t) 2). It is well-known that when ρ(s) = s p 1 s with p > 1, E(t) decays as t 2/(p 1) and this decay rate is optimal: the result is due to Haraux Idea of the proof: E is a Lyapunov function, but not a strict one, one misses estimates for potential energy Introduce the function, which is indeed a modified energy V ε (t) = E(t) + ε u(t) p 1 u(t)u (t)

38 The scalar case We shall prove that it is a Lyapunov function, which is equivalent to the original one for sufficiently small ε, uniformly with respect to t. Let us first check that V ε is equivalent to E as above stated. Noting that since E is non increasing, we have u(t) 2 2ν 1 E(t) 2ν 1 E(0) for all t 0, so that V ε (t) E(t) ε(2ν 1 p 1 u(t) 2 E(0)) + ε u (t) ε max(2ν 1 E(0)) p 1, 1)E(t) εc E(0) E(t) t 0 where C E(0) > 1. Therefore we have (1 C E(0) ε)e(t) V ε (t) (1 C E(0) ε)e(t) t 0 Hence V ε is equivalent to E for sufficiently small ε, uniformly with respect to t.

39 The scalar case The additional term ε u(t) p 1 u(t)u (t) is build to "create" a potential term which is negative when one differentiates it with respect to t, so that the modified energy is a strict Lyapunov function Let us precise what it means (not with all the details but just the idea):

40 The scalar case When one differentiates V ε, two good terms appear: u (t) p+1 which is due to the contribution E and thus to the dissipation relation εν u(t) p+1 which is the important term which appears in the expression ε u p 1 uu The other terms can be absorbed in this two dominant terms, so that V ε(t) c 1 u (t) p+1 εc 2 u(t) p+1 where c 1 > 0, c 2 > 0. Hence we have where c > 0. V ε(t) cv ε (p+1)/2 (t)

41 The scalar case Thus since p > 1, we deduce that (V (1 p)/2) ε ) (t) c(p 1)/2 > 0 so that for ε sufficiently small, and t sufficiently large ( V ε (t) V (1 p)/2 ε ) 2/(p 1) (0) + c 3 t CE(0) t 2/(p 1) so that since V ε E uniformly in t for ε sufficiently small, we have for t sufficiently large E(t) C E(0) t 2/(p 1)

42 The scalar case an alternative proof based on integral inequalities Haraux 1978, Komornik Indeed one can show that there exists C 0 > 0 (not depending on E 0 such that T S E (p 1)/2 (t)e(t) dt C 0 E(S) 0 S T. where p > 1. One can show that if E is a nonnegative, non increasing, absolutely continuous function satisfying the above inequality then there exists C E( 0) such that Remark E(t) C E(0) t 2/(p 1) for sufficiently large t. If p = 1 in the above integral inequality, then one can show that E is exponentially decaying at infinity.

43 The scalar case Let us give a proof for p = 1 (linear damping case), that E is exponentially decaying Set (this is well-defined thanks to our integral inequality) k(t) = From our assumption, we have Differentiating k, we get t E(s) ds for t 0 k(t) C 0 E(t) t 0 Thus we have k (t) = E(t) k(t)/c 0 t 0 k(t) k(0)e t/c 0 C 0 E(0)e t/c 0 t 0

44 The scalar case Using non negativity of E and its decaying property we have C 0 E(t) t t C 0 E(s) ds k(t C 0 ) C 0 E(0)e (t C 0)/C 0 t 0, We finally get E(t) E(0)e 1 t/c 0 t 0

45 The scalar case For p > 1, we also write a differential inequality for a suitable function k p, more precisely for k p (t) = the differential inequality being t E (p+1)/2 (s) ds k p(t) = E (p+1)/2 (t) C (p+1)/2 0 k (p+1)/2 p (t) t 0 One can get the announced power-like decay as for its proof for the Lyapunov function V ε.

46 The scalar case What about proving a lower estimate for the energy? there exists c > 0 such that E = u p+1 ce (p+1)/2 = after integration (E (1 p)/2 ) c = E(t) c 1 t 2/(p 1) for sufficiently large t. So this proves the optimality of the upper estimate.

47 The scalar case Can this type of results be extended for general feedbacks, i.e. with "arbitrary" growth close to 0? How to build an appropriate a suitable weight for integral inequalities for general feedbacks? How to deduce a simple computable optimal decay rate from the weighted integral inequality if we get one? And further can it be extended to infinite dimensional systems, nonlocal dissipation...?

48 The optimal-weight convexity method Sommaire 1 General motivations for dissipative vibrating systems 2 Dissipative systems in finite dimensions 3 Infinite dimensional dissipative systems 4 The scalar case 5 The optimal-weight convexity method 6 Lower energy estimates: an energy comparison principle 7 Examples of decay rates and optimality 8 Vectorial case: parameter-dependent upper estimates

49 The optimal-weight convexity method Assume that the function g that measures the behavior of the feedback ρ close to 0 is an odd, strictly increasing smooth function on R. We assume that g (0) = 0, otherwise it is easy to show that E decays exponentially to 0 at (linear damping case). We set H(s) = sg( s) s 0

50 The optimal-weight convexity method Weighted integral inequalities Theorem (A.-B. 2005, 2010) We make the above hypotheses and further assume that H is strictly convex on a right neighborhood of 0, denoted by [0, r 2 0 ], r 0 > 0. Then there is a nonnegative smooth strictly increasing weight function w such that E satisfies the following integral inequality T S w(e)e dt C 0 E(S) 0 S T, Proof Set θ = min ( 1, 1 ). 2 µ

51 The optimal-weight convexity method Let for the moment w be a nonnegative C 1 and strictly increasing function defined from [0, r0 2 ) onto [0, + ). w is going to be an optimal-weight function to be determined later on We multiply the left hand side of the equation u + νu + f (u) + ρ(u ) = 0 by w(e(t))u(t) and integrate the resulting equation on [S, T ]. Since E is nonincreasing, w is nondecreasing and thanks to our assumption on f, this gives

52 The optimal-weight convexity method continued. T T θ Ew(E) dt S S 1 T 2 νθ 4 T S T S S w(e) u 2 dt 1 2 w (E) E u u dt 1 2 w(e) u 2 dt + 1 4νθ T S T S w(e)ρ(u )u+ [ w(e)u u ] T S w(e) ρ(u ) 2 dt+ w(e) u 2 dt + 1 ν E(S)w(E(S)), 0 S T. where θ is defined as above.

53 The optimal-weight convexity method continued. Thus, we have T S Ew(E) dt 2 θ 1 2νθ 2 T S T S w(e) u 2 dt+ w(e) ρ(u ) 2 dt + 2 νθ E(S)w(E(S)), 0 S T.

54 The optimal-weight convexity method The next steps will be devoted to control the weighted integrals of the linear kinetic and nonlinear kinetic energies in the right hand side of the above inequality.

55 The optimal-weight convexity method We first remark that thanks to our hypotheses on ρ, we have (up to the positive constants c and C, which may change) { c s ρ(s) C s, s r 0, c g( s ) ρ(s) C g 1 ( s ), s r 0, We set Ĥ(x) = { H(x), if x [0, r 2 0 ], +, if x R [0, r 2 0 ], We define Ĥ as the convex conjugate function of Ĥ (also called the Legendre transform), i.e. Ĥ (y) = sup(xy Ĥ(x)) We recall that the following inequality, called Young s inequality AB Ĥ(A) + Ĥ (B) A 0, B 0

56 The optimal-weight convexity method First step: estimate of the linear kinetic energy For u r 0, we have H( u 2 ) = u g( u ) 1 c u ρ(u ). This, together with Young s inequality imply for u r 0 w(e) u 2 Ĥ ( w(e) ) + H( u 2 ) Ĥ ( w(e) ) + 1 c u ρ(u ). On the other hand, we have w(e) u 2 1 c w(e)u ρ(u ), for u r 0.

57 The optimal-weight convexity method continued. Combining the above two inequalities and the dissipation relation, we obtain T S w(e) u 2 dt T S Ĥ ( w(e) ) dt + 1 c [ 1 + w(e(s)) ] E(S), 0 S T.

58 The optimal-weight convexity method continued. Second step 2: estimate of the nonlinear kinetic energy For u r 0, we have, thanks to Young s inequality w(e) ρ(u ) 2 C 2 Ĥ ( w(e) ) + ( ρ(u ) H 2 ) C Ĥ ( w(e) ) + 1 C u ρ(u ), for u r 0. On the other hand, we have w(e) ρ(u ) 2 Cw(E)u ρ(u ), for u r 0.

59 The optimal-weight convexity method continued. Combining the above two inequalities and the dissipation relations, as above, we have T S w(e) ρ(u ) 2 dt T C 2 Ĥ ( w(e) ) dt + C [ 1 + w(e(s)) ] E(S), 0 S T. S Using the above estimates, we obtain the estimate

60 The optimal-weight convexity method Proof. (continued) T S T Ew(E) dt β Ĥ ( w(e) ) dt+ S 2 ( + 2 νθ θc + C 2νθ )[ 1 + w(e(s)) ] E(S), 0 S T, where β can be easily computed and does not depend on w. One has to be cautious, β is not only chosen as the constant appearing in the above right hand side, it also should be chosen in relation with E(0) such that the weight function s w(s) is defined in the range [0, E(0)) (indeed we choose it even with a stronger criterium below for technical reasons).

61 The optimal-weight convexity method continued. We can now choose the weight by requesting We define a function L by Then the weight w should satisfy βĥ ( w(e) ) = 1 2 Ew(E) L(y) = Ĥ (y) y that is, if L is invertible L(w(E)) = E/(2β) w(.) = L 1 (./2β).

62 The optimal-weight convexity method continued. Assume for the moment that w is well-defined and satisfies the desired properties (non negativity, strictly increasing...). We saw before that the energy E satisfies T S T Ew(E) dt β Ĥ ( w(e) ) dt+ S 2 ( + 2 νθ θc + C 2νθ )[ 1 + w(e(s)) ] E(S), 0 S T, With this choice of weight function, we deduce that T S Ew(E) dt C 0 E(S), 0 S T where C 0 > 0, so that we prove that E satisfies a generalized weighted inequality.

63 The optimal-weight convexity method We now check that the optimal-weight w is indeed well-defined by this way Proposition (A.-B. 2005) Let g be a given odd, strictly increasing C 1 function from R to R such that g (0) = 0. We assume that there exists r 0 > 0 such that the function H is strictly convex on [0, r0 2 ]. Then the function L defined by Ĥ (y), if y (0, + ), L(y) = y 0, if y = 0, is the strictly increasing continuous onto function from [0, + ) on [0, r0 2 ) given by:

64 The optimal-weight convexity method Proposition (continued) (H ) 1 (y) H((H ) 1 (y)), if y [0, H (r0 2 )], L(y) = y r0 2 H(r 0 2), if y [H (r0 2 ), + ). y The weight function w is thus uniquely determined as w(s) = L 1 ( s 2β ) s [0, 2βr 2 0 ). where β > 0 is a suitable constant (independent on w) which satisfies β > E(0)/(2r 2 0 ). Remark Note that for general dampings the inverse of L is not defined on all R, and the further requested computations are not explicit, contrarily to the linear or polynomial case.

65 The optimal-weight convexity method Upper energy estimates from the weighted integral inequality Hence E is a nonnegative, nonincreasing continuous function satisfying a weighted integral inequality. In the polynomial case, it leads to a polynomial decay rate. The situation is more tricky in the general case. We still assume that H is strictly convex on [0, r0 2 ]. Let us introduce the following important function (it will be also useful to classify the feedbacks and for optimality) Λ H on (0, r0 2 ] defined by Λ H (x) = H(x) xh (x).

66 The optimal-weight convexity method Theorem (A.-B. JDE 2010) Let H be a given strictly convex C 1 function from [0, r0 2 ] to R such that H(0) = H (0) = 0, where r 0 > 0 is sufficiently small. We define Ĥ, L and Λ H as before. Assume E be a given nonincreasing, absolutely continuous, nonnegative real function defined on [0, + ), C 0 > 0 be a fixed real number and β > 0 a given real number such that E satisfies the nonlinear Gronwall inequality T S under the condition E(t)L 1 ( E(t) 2β ) dt C 0E(S), 0 S T. 0 < E(0) 2L(H (r 2 0 )) β,

67 The optimal-weight convexity method Theorem (continued) Then, if lim sup x 0 + Λ H (x) < 1, E decays at infinity as follows: E(t) 2β(H ) 1( DC 0 ), for sufficiently large t t where D is a positive constant which does not depend on E(0). Otherwise, we have a general decay rate (see A.-B. 2005). Remark If g(s) = s p 1 s, then H(s) = s (p+1)/2 so that H (s) = 1 (p + 1)s(p 1)/2 2 Thus we recover the upper estimate E(t) C(E(0))t 2/(p 1).

68 The optimal-weight convexity method Moreover one can show that the weight function is of the form w(s) = Cs (p 1)/2 that is we recover as a peculiar example the weight introduced by Haraux-Komornik. If g is such that lim sup Λ H = 1, this means that the feedback is "close" to a linear feedback as for instance g(x) = x(ln( 1 x )) p where p > 0, x close to 0 + We will come back later on this criterium. Under this criterium, we have a simple upper estimate. Next questions: Determine a sharp lower estimate Optimality of the upper estimate?

69 The optimal-weight convexity method Moreover one can show that the weight function is of the form w(s) = Cs (p 1)/2 that is we recover as a peculiar example the weight introduced by Haraux-Komornik. If g is such that lim sup Λ H = 1, this means that the feedback is "close" to a linear feedback as for instance g(x) = x(ln( 1 x )) p where p > 0, x close to 0 + We will come back later on this criterium. Under this criterium, we have a simple upper estimate. Next questions: Determine a sharp lower estimate Optimality of the upper estimate?

70 The optimal-weight convexity method Moreover one can show that the weight function is of the form w(s) = Cs (p 1)/2 that is we recover as a peculiar example the weight introduced by Haraux-Komornik. If g is such that lim sup Λ H = 1, this means that the feedback is "close" to a linear feedback as for instance g(x) = x(ln( 1 x )) p where p > 0, x close to 0 + We will come back later on this criterium. Under this criterium, we have a simple upper estimate. Next questions: Determine a sharp lower estimate Optimality of the upper estimate?

71 Lower energy estimates: an energy comparison principle Sommaire 1 General motivations for dissipative vibrating systems 2 Dissipative systems in finite dimensions 3 Infinite dimensional dissipative systems 4 The scalar case 5 The optimal-weight convexity method 6 Lower energy estimates: an energy comparison principle 7 Examples of decay rates and optimality 8 Vectorial case: parameter-dependent upper estimates

72 Lower energy estimates: an energy comparison principle Lower energy estimates: energy comparison principle We prove Lemma (A.-B. JDE 2010) Assume that f is a continuous and locally Lipschitz function on R which satisfies the above assumptions, and that ρ = g satisfies (A1). Moreover assume that H is increasing and H(0) = 0. Let u be a solution of the scalar ode and E be its energy. Then the following lower estimate holds 1 2 v 2 (t) E(t), t 0, where v is the solution the ODE v + g(v) = 0, v(0) = 2E(0).

73 Lower energy estimates: an energy comparison principle Proof. Thanks to the dissipation relation, and to our assumptions on g, we have E (t) u (t)g(u (t)) = H ( (u ) 2), t 0. Hence, thanks to the ode satisfied by v, we have ( v 2 2 E ) (t) = H ( (u ) 2) H ( (v(t)) 2) H(2E(t)) H ( (v(t)) 2), t 0, and v 2 (0) = 2E(0).

74 Lower energy estimates: an energy comparison principle Proof. Since H is strictly increasing on R, we deduce easily by comparison principles for ODE s that the stated lower energy estimate holds. This an energy comparison principle: we compare the energy of the nonlinear harmonic oscillator u + ν u + f (u) + ρ(u ) = 0. which is a second order ODE, to the energy of the first order ODE v + g(v) = 0

75 Lower energy estimates: an energy comparison principle Theorem (Optimality Theorem, A.-B. JDE 2010) Assume that f is a continuous and locally Lipschitz function on R which satisfies the above hypotheses, and that ρ = g satisfies (A1). We assume that H is strictly convex on [0, r 2 0 ]. Let (u 0, u 1 ) R 2, satisfying 0 < u 1 + u 0 be given, u be the corresponding solution and E be its energy. Moreover assume that either 0 < lim inf x 0 or that there exists µ > 0 such that ( H(µ x) z1 0 < lim inf x 0 µ x x Λ H(x) lim sup Λ H (x) < 1, x 0 1 ) H(y) dy for a certain z 1 (0, z 0 ] (arbitrary)., and lim sup Λ H (x) < 1, x 0

76 Lower energy estimates: an energy comparison principle Theorem (continued) Then the energy of solution satisfies the estimate E(t) = O(v 2 (t)) = O ((H ) 1 ( 1 ) t ), uniformly for large time The proof relies on a key comparison lemma that relies on convexity properties and allows up to compare the lower estimate v 2 (t) which is an "energy-type" estimate, since v 2 is the energy associated to the ode v + g(v) to the upper time-pointwise estimate (H ) 1 (Cste/t)

77 Lower energy estimates: an energy comparison principle Remarks The function Λ H introduces a "classification" of the nonlinearity of the feedbacks lim x 0 Λ H (x) = 0 for g(x) = e 1/x, x > 0 and more generally for very degenerate feedbacks (converging to 0 exponentially for instance) lim x 0 Λ H = 2/(p + 1) for g(x) = x p 1 x and more generally this lim inf (0, 1) for polynomial-logartithmic behavior close to 0 lim x 0 Λ H (x) = 1 for g(x) = x ln(x) 1, x > 0 and more generally for feedbacks which "are close" to a linear behavior at the origin. for feedbacks for which this lim sup < 1, all the solutions have the same asymptotic behavior at infinity

78 Lower energy estimates: an energy comparison principle The condition limsup x 0 +Λ H (x) < 1 for optimality excludes the feedbacks which are "close" to a linear behavior at the origin (see the 3rd example above). Is this sharp? Consider a linear case, that is u + ν u + υu = 0. where ν > 0, υ > 0 are given constants and u is a scalar unknown. Then if υ 2 4ν > 0 there are two linearly independent solutions, namely u 1 (t) = e υ υ 2 4ν 2, t 0, and u 2 (t) = e υ+ υ 2 4ν 2, t 0,

79 Lower energy estimates: an energy comparison principle The energies E i (t) = 1 2 ( u i (t) 2 + ν u i (t) 2), i = 1, 2, t 0, of these two solutions decay exponentially as t goes to, but at different rates and their ratio satisfies E 2 (t) lim t E 1 t) =. This implies different lower and upper bounds and different decay rates depending on the initial data Same behavior for υ 2 4ν = 0, If υ 2 4ν < 0, all the energies of all the solutions decay at the same speed e υt as t goes to. A similar situation (with more cases) arises in the linear vectorial case.

80 Lower energy estimates: an energy comparison principle We conjecture that a similar situation, that is two branches of solutions with different asymptotic behavior may arise for nonlinear feedbacks close to linear growth at the origin that is for feedbacks for which. limsup x 0 +Λ H (x) = 1

81 Examples of decay rates and optimality Sommaire 1 General motivations for dissipative vibrating systems 2 Dissipative systems in finite dimensions 3 Infinite dimensional dissipative systems 4 The scalar case 5 The optimal-weight convexity method 6 Lower energy estimates: an energy comparison principle 7 Examples of decay rates and optimality 8 Vectorial case: parameter-dependent upper estimates

82 Examples of decay rates and optimality Examples of decay rates Example 1 (polynomial case): let g be given by g(x) = x p where p > 1 on (0, r 0 ]. Then E(t) Cβ E(0) t 2 p 1, (3) for t sufficiently large and for all (u 0, u 1 ) in R 2. Moreover this estimate is optimal. Example 2 (exponential case): let g be given by g(x) = e 1 x 2 on (0, r 0 ]. Then E(t) Cβ E(0) (ln(t)) 1, for large t. Moreover, this estimate is optimal.

83 Examples of decay rates and optimality Example 3 (polynomial-logarithmic, close to linear): let g be given by g(x) = x(ln( 1 x )) p where p > 0. Then E(t) C β E(0) e 2( p t DT 0 ) 1/(p+1) t 1/(p+1) (4) for t sufficiently large. Optimality cannot be asserted. Example 4 (faster than any polynomial less than exponential) : vskip 2mm let g be given by g(x) = e (ln( 1 x ))p, 1 < p < 2, x [0, r 0 ]. Then E(t) C β E(0) e 2(ln(t))1/p This estimate is optimal.

84 Vectorial case: parameter-dependent upper estimates Sommaire 1 General motivations for dissipative vibrating systems 2 Dissipative systems in finite dimensions 3 Infinite dimensional dissipative systems 4 The scalar case 5 The optimal-weight convexity method 6 Lower energy estimates: an energy comparison principle 7 Examples of decay rates and optimality 8 Vectorial case: parameter-dependent upper estimates

85 Vectorial case: parameter-dependent upper estimates Similar result with sharp energy decay rates, and optimality results for the vectorial case. These optimal estimates depends on the dimension of the system. Moreover these results applied to semi-discretized PDE s such as the wave or plate equations and give optimal rates of decay. Therefore when applied to semi-discretized PDE s, they are not uniform with respect to the discretization parameter.

86 Vectorial case: parameter-dependent upper estimates We consider the vectorial case: u + Au + f (u) + Bρ(u ) = 0. where the unknown u R n and A is a symmetric, positive definite matrix B = diag(b i ) 1 i n, with b i 0 i {1,..., n} (f (u)) i = f (u i ), (ρ(u )) i = ρ(u i ) ρ : v = (v 1,..., v n ) R n ( g 1 (v 1 ),..., g n (v n ) ). f is a vectorial function of the form: f (u) = (f 1 (u 1 ),..., f n (u n )), u = (u 1,..., u n ) R n.

87 Vectorial case: parameter-dependent upper estimates Recall that the energy is given by E(t) = 1 2 ( u 2 + Au, u + n ) (F(u i )), i=1 and the dissipation relation by n E (t) = b i ρ(u i )u i 0 t 0. i=1

88 Vectorial case: parameter-dependent upper estimates Theorem (A.-B. 2010) Assume that f is continuous and locally Lipschitz on R n and satisfies some growth conditions (as for the scalar case) the functions g i (components of the feedback ρ) satisfies similar assumptions than in the optimality theorem for the scalar case with cg( s ) g i (s) k 1 g( s ) and g(s) k 2 g 1 ( s ) for s 1 and i {1,..., n} the function H(s) = sg( s) is strictly convex on [0, r0 2]

89 Vectorial case: parameter-dependent upper estimates Theorem (continued) Then If 0 < lim inf x 0 then the following estimates hold Λ H(x) lim sup Λ H (x) < 1, x v 2 (t) E(t) C 2 v 2 (t), for t sufficiently large, where C 2 is a positive constant and v is the solution of the ordinary differential equation: v (t) + n k 1 g(v(t)) = 0, v(0) = 2E(0), t 0. Or equivalently, we have E(t) = O(v 2 (t)) = O ((H ) 1 ( D ) t ) uniformly with respect to t, (5) where D is a positive constant.

90 Vectorial case: parameter-dependent upper estimates Theorem (continued) If there exists µ > 0 and z 1 (0, E(0)] such that ( H(µ x) z1 0 < lim inf x 0 µ x x 1 ) H(y) dy, and lim sup Λ H (x) < 1, x 0 then the above estimates hold.

91 Vectorial case: parameter-dependent upper estimates From the above theorem, we deduce easily the following corollary. Theorem (Semi-discretized PDE s) Assume that the vectorial functions defined by f i (.) = f (x i,.) and ρ(.) = ρ(x i,.)for i = 1,... n satisfy the above hypotheses, where the points x i, i = 1,..., n denotes the discretization points. Then the above estimates hold. Remark The decay rate given in the above theorem depends on n and thus on the discretization parameter h. They are not uniform in h. In particular, for prescribed initial data, the constant β behaves as C/h where C depends on the initial data but not on h. Upper estimates independent on h and sharper conditions on the nonlinearity: work in progress with Emmanuel Trélat and Yannick Privat (see course by Emmanuel).

92 Vectorial case: parameter-dependent upper estimates Hence in the finite dimensional case, we have a simple and optimal energy decay rate for finite dimensional systems, without prescribed growth assumptions on the feedbacks it applies in particular to finite dimensional systems coming out from semi-discretization of infinite dimensional nonlinear vibrating systems. this, for fixed discretization parameters, since our estimates in the vectorial case depend on n, i.e. equivalently on h. see Emmanuel s talk for extension to non diagonal matrix B (related to discretized nonlinear damping) Next lecture: infinite dimensional systems