From averaged to simultaneous controllability of parameter dependent finite-dimensional systems

Transcription

1 From averaged to simultaneous controllability of parameter dependent finite-dimensional systems Jérôme Lohéac a and Enrique Zuazua b,c a LNAM niversité, IRCCyN MR CNRS 6597 Institut de Recherche en Communications et Cybernétique de Nantes, École des Mines de Nantes, 4 rue Alfred Kastler, 4437 Nantes, France b BCAM Basque Center for Applied Mathematics, Alameda rquijo 36-5, Plaza Bizkaia, E-48, Bilbao, Basque Country, Spain c Ikerbasque Basque Foundation for Science, María Díaz de Haro 3, 483 Bilbao, Basque Country, Spain May 5, 5 Abstract We consider a linear finite dimensional control system depending on unknown parameters. We aim to design controls, independent of the parameters, to control the system in some optimal sense. We discuss the notions of averaged control, according to which one aims to control only the average of the states with respect to the unknown parameters, and the notion of simultaneous control in which the goal is to control the system for all values of these parameters. We show how these notions are connected through a penalization process. Roughly, averaged control is a relaxed version of the simultaneous control property, in which the differences of the states with respect to the various parameters are left free, while simultaneous control can be achieved by reinforcing the averaged control property by penalizing these differences. We show however that these two notions require of different rank conditions on the matrices determining the dynamics and the control. When the stronger conditions for simultaneous control are fulfilled, one can obtain the later as a limit, through this penalization process, out of the averaged control property. AMS subject classification: Key words: Controllability, parameter dependent system, averaged control, simultaneous control, penalization. This work was supported by the Advanced Grants NMERIWAVES/FP of the European Research Council Executive Agency, FA of the EOARD-AFOSR, PI-4 and the BERC 4-7 program of the Basque Government, the MTM-936-C- and SEV-3-33 Grants of the MINECO. The first author thanks BCAM for its hospitality and support as a Visiting fellow of the NMERIWAVES Advanced Grant of the ERC. The authors thank the CIMI - Toulouse for the hospitality and support during the preparation of this work in the context of the Excellence Chair in PDE, Control and Numerics. addresses: Jerome.Loheac@irccyn.ec-nantes.fr J. Lohéac, ezuazua@bcamath.org E. Zuazua.

2 Introduction We consider a parameter dependent control system: ẏ = A y + B u t, T,.a y = y i..b In order to fix the notation, all along this paper, is a random parameter the system s parameter following a probability law µ, with, F, µ a probability space, = R n is the state space and = R m the control one. We assume that for every, A L and B L,. The control t ut is assumed to be independent of the parameter whereas the state y t = y t; u is time and parameter dependent. In addition, by Duhamel formula, y can be represented as follows: T y t; u = e ta y i + e t sa B us ds, t, u L loc R +,.. Let us also define the space: L, ; µ = { y, } y dµ,.3 which is an Hilbert space endowed with the scalar product: y, z L,;µ = y, z dµ y, z L, ; µ. In section we introduce precise conditions on A, B ensuring that for every t and every u L loc R +,, y t; u L, ; µ whenever the parameter-dependent initial data satisfy y i L, ; µ. This paper is devoted to analyse the following controllability problems. Averaged controllability: The system is said to be averaged controllable in time T > if, for every y i L, ; µ and every y f, there exists u L [, T ], such that: y T ; u dµ = y f,.4 In other words, averaged controllability is the control of the expectation of the system s output. This notion is illustrated on Figure a. Exact simultaneous controllability: The system is said to be exactly simultaneously controllable in time T > if, for every y i, y f L, ; µ, there exists u L [, T ], such that: This notion is illustrated on Figure b. y T ; u = y f µ a.e.,.5

3 Approximate simultaneous controllability: The system is said to be approximately simultaneously controllable in time T > if, for every y i, y f L, ; µ and every ε >, there exists u L [, T ], such that: y T ; u y f dµ ε..6 This notion is illustrated on Figure c. average trajectory y f y f y f y i y i parameter dependent trajectories y i parameter dependent trajectories parameter dependent trajectories a Averaged controllability. b Simultaneous controllability. c Approximate Simultaneous controllability. Figure : Different controllability notions, introduced in.4,.5 and.6, for parameter dependent systems, with initial condition and target independent of. Remark... Even if the system. is controllable in average, this fact does not give any information on the variance of the outputs.. It is obvious that the exact simultaneous controllability property implies the averaged controllability and the approximate simultaneous controllability ones. In addition, one can find systems which are controllable in average resp. approximatively simultaneously controllable which are not exactly simultaneously controllable. Moreover, the approximate simultaneous controllability property implies the averaged controllability one. In fact, the approximate simultaneous controllability property ensures that given T >, y i L, ; µ, y f and ε >, there exists u ε L [, T ], such that But, by Cauchy-Schwarz inequality, y T ; u ε y f ε. L,;µ y T ; u ε y f dµ y T ; u ε y f dµ. Thus, the system is approximatively controllable in average i.e. the linear and continuous map Φ : u L [, T ], T et ta B ut dtdµ has a dense image in. But since is a finite dimensional vector space, we obtain Im Φ =, i.e. the system is controllable in average. 3

4 3. There is no natural ordinary differential equation describing the average Y t = y t dµ, except when A is independent of for which we have: Ẏ = AY + B dµ u. In this particular case, the averaged controllability property is equivalent to the controllability of the pair A, B dµ. 4. When = {,, K } is of finite cardinal, the simultaneous controllability is equivalent to the classical controllability one for the augmented system: ẏ = Ay + Bu, with: y = y. A, A =... and B = B.. y K A K And the controllability of this system is equivalent to the Kalman rank condition: [ ] rank B AB A Kdim B = K dim. B K 5. In the previous item, we have seen that the simultaneous controllability property when the cardinal of is finite can be interpreted in terms of a classical rank condition. But, when is infinite, the output of the system is the function y T, living in an infinite-dimensional space. The first issue to be addressed is the choice of the norm in that space. In the following, we choose the L -norm. Accordingly, the fact that y T = y f holds for almost every with respect to the measure µ is guaranteed by the fact that y T y f dµ =. This choice is natural, since in the particular case where y f = y f is independent of and y T dµ = y f, the integral y T y f dµ is the variance of the system s output. Thus, the L -norm approach is natural from a probabilistic point of view but one could also use any L p, ; µ-norm. In the next item, we mention some existing literature when considering the L -norm. 6. For parameter dependent systems, J.-S. Li and N. Khaneja [5] see also J.-S. Li [4] introduced the notion of ensemble controllability: The system is said to be ensemble controllable in time T > if, for every ε and every y i, yf, there exists u L [, T ], such that: y T ; u y f ε..7 This notion of ensemble controllability, which does not seem to have a probabilistic interpretation, is similar to our notion of approximate simultaneous controllability above, where the L, ; µ-norm is replaced by the L, one. Controlling the average or the expectation of a parameter dependent system is not a new problem. It has been previously studied when a classical control system is perturbed by an additional drift V. A. grinovskii [3], A. V. Savkin and I. R. Petersen [], I. R. Petersen [9]. We present here a different frame for which the uncertainty is inside the system itself, and not due to some external noise. Taking 4

5 into account that we only know the probability distribution of the unknown parameter, it is natural to try to control the expectation of the output of the system. In [4], it has been shown that the averaged controllability property is equivalent to a Kalman rank condition of infinite order. However, even if the average of the system is controlled, this fact does not ensure that the output of system is close to the desired target for any specific realisation of the parameter. Of course, the ideal situation arises when all the parameter dependent trajectories exactly reach the desired target. This corresponds, precisely, to the notion of simultaneous controllability. Classically, the simultaneous exact controllability property corresponds, by duality, to the one of simultaneous exact observability see 3.. However, when is an infinite dimensional set, those properties are difficult to check in practice. This is why, in this article, we show that, if the simultaneous controllability property holds, then the approximate simultaneous control can be achieved from the averaged controls by means of a penalisation procedure and at the limit, when the penalizing parameter goes to, we recover the simultaneous control. The notion of simultaneous controllability was introduced by D. L. Russell [] see also J.-L. Lions [6, Chapter 5] for partial differential equations. As mentioned above, when dealing with finite dimensional systems and when the parameter ranges over a finite set, the problem can be handled through classical rank conditions. However, the issue is much more complex when the parameter ranges over an infinite set. The averaged controllability property has already been tackled by E. Zuazua et al [4, 3, 8] for some relevant PDE models. However, the link between the averaged and simultaneous controllability in that setting has not been yet developed. The tools developed here could be used to handle PDE and, in general, infinite-dimensional systems, but this requires further efforts. In general, the simultaneous controllability problem is set in an infinite dimensional space this holds when the cardinal of is infinite. In infinite dimensional spaces the choice of the norm is important and an appropriate choice has to be done. According to the 5 th item of Remark., we chose the weighted L -norm, that corresponds to the variance. More precisely, the simultaneous controllability property.5 holds if: y T y f dµ =..8 Consequently, in section 4, we introduce the parametrized optimal control problems: min J κ u = T ut dt + κ y T ; u y f dµ y T ; u dµ = y f dµ κ, with y the solution of. with control u and initial condition y i. We will see in Theorem 4. that, at the limit κ, the minimum u κ is a control which minimizes the variance of the system s outputs. For instance, we will see that if the sequence J κ u κ κ is bounded then the sequence u κ κ converges to a control u which solves the minimisation problem: min T ut dt y T ; u y f dµ =. 5

6 In other words, u is the HM control the control obtained from the Hilbert niqueness Method for the simultaneous control problem. More generally, the result of Theorem 4. can be summarized in Table, where we have defined y L, ; µ as the minimizer of y y f L,;µ under the constraints y dµ = y f dµ and y {y T ; u, u L [, T ], }. uκ L κ bounded unbounded y T ; u κ y f L,;µ κ converge to do not converge to simultaneous exact controllability simultaneous approximate controllability simultaneous exact controllability to y simultaneous approximate controllability to y Table : Possible behaviors as κ, y beeing defined by 4.4. This penalty argument is natural and has already been used in control theory. In J.-L. Lions [7] it was used to achieve approximate controllability as the limit of a sequence of optimal control problems see also L. A. Fernández and E. Zuazua [] for semi-linear heat equations. This penalty method has also been used numerically, for the numerical approximation of null controls for parabolic problems see F. Boyer []. This paper is organized as follows. In section, we give some conditions on A, B and y i ensuring that the problem we are considering is well defined. Then, in section 3, we recall some known results about averaged controllability and we describe the duality approach for simultaneous controllability. In section 4, we present the penalty method and give some convergence results. More precisely, in this section we prove the main theorem Theorem 4. of this article. Then, in section 5, we present some results for a further numerical development of the case where is a countable set. Finally, in section 6, we conclude this work by some general remarks and open problems. Admissibility conditions In this section, we give some conditions ensuring that y t dµ and y t dµ are well defined. Let us consider the Hilbert space L, ; µ defined by.3. sing Cauchy-Schwarz together with dµ =, leads to: y dµ y L,;µ y L, ; µ. Thus, in this paragraph, we only give conditions on A, B and µ such that y t L,;µ < and in all this article, we assume that initial and final condition are elements of L, ; µ. By Duhamel formula, the solution y t = y t; u of. is given by., i.e., t y t; u = e ta y i + e t sa B us ds, t. 6

7 Lemma.. Set A L. For every T > and every, there exists ς T > such that: e T A e T A y ς T y y. Assume: ς T < µ a.e... Then for every T >, there exists ςt > ςt = sup ς T such that: e T A y i L,;µ ςt yi L,;µ y i L, ; µ.. Proof. The existence of ς T is clear. The result follows from Cauchy-Schwarz inequality. Example.. Let, F, µ be a probability space. If for every, A is skew-adjoint, then. holds with ςt =. In this case, we have ς T = for every. Lemma.. Set A L and B L,. For every T > and every, there exists a constant C T > such that: Assume that: T e T ta B ut dt Then for every T >, there exists CT > such that: T e T ta B ut dt C T u L [,T ],. C T dµ <..3 dµ CT u L [,T ], u L [, T ],. Proof. The existence of C T > independent of u is classical. The result follows from Minkowski and Cauchy-Schwarz inequalities. Thus, if A and B satisfies the assumption of lemmas. and., then for every y i L, ; µ, yt ; u defined by. is an element of L, ; µ. From these two lemmas, we can derive the following corollaries: Corollary.. Assume Card < and set A, B L L,, then for every y i L, ; µ, and every u L loc R +,, the solution y t; u of. belongs to L, ; µ for every t. Corollary.. Assume R d is a bounded set and assume the map A, B is continuous on co, with co the smallest convex set containing. Then for every y i L, ; µ, every u L loc R +, and every t, the solution y t; u of. belongs to L, ; µ. 7

8 Proof. Since and are finite dimensional spaces, for every co, ς T := sup e T A e T A y and C T := sup y, y = u L [,T ],, u L [,T ], = T e T ta B ut dt are well defined for every co and every T. Moreover, since co A, B L L, is continuous, the map co ς T, C T R is continuous, thus bounded. The result follows from lemmas. and.. Remark.. Even if Corollary. can be proved directly, it can also be seen as a consequence of Corollary.. Corollary.3. Let, F, µ be a probability space and assume A skew-adjoint for every. If B L, dµ <, then for every y i L, ; µ and every u L loc R +,, the solution y t; u of. belongs to L, ; µ for every t. Proof. According to Lemma. and Example., we have have: t e t sa B us ds t e t sa B us ds = Thus the assumptions of Lemma. are fulfilled. 3 Duality approach and Kalman rank conditions t e ta y i L, ; µ. In addition, we B us ds t B L, u L [,t],. Here and in the sequel we assume that the hypotheses of lemmas. and. are satisfied. 3. State of the art for averaged controllability Let us recall some known results on averaged controllability for finite dimensional systems. These results are taken from [4]. Theorem 3. [4] Theorem. System. fulfills the averaged controllability property.4 if and only if the following rank condition is satisfied: [ ] rank A j B dµ, j = dim. 3. This result is based on duality arguments. More precisely, we introduce the parameter dependent adjoint system: z = A z t, T, 3.a z T = z f. 3.b Notice that even if this system depends of the parameter the final condition z f is independent of. The next result makes the link between averaged controllability, and averaged observability and gives also a link between the adjoint system and the control of minimal L -norm. 8

9 Theorem 3. [4] Theorem. System. fulfills the averaged controllability property.4 if and only if the adjoint system 3. satisfies the averaged observability inequality: T ct z f B z t dµ dt z f, 3.3 where ct > is a constant independent of z f. In addition, both conditions are equivalent to the rank condition 3.. When these properties hold, the averaged control of minimal L [, T ], -norm is given by: ut = B z t dµ t, T, 3.4 where { z } is the solution of the adjoint system 3. corresponding to the datum z f minimizing the functional: J : R z f T B z t dµ dt y f, z f + y i, z dµ Observability inequality for exact simultaneous controllability Let us define for every the adjoint system of.: z = A z t, T, 3.6a z T = z f. If the system. is simultaneously controllable then for every z f L, ; µ, y T y f, z f =. L,;µ That is to say: T ut, B z t dµ dt = Let us then define the cost function J by: J : L, ; µ R z f T y f, z f y L,;µ i, z. L,;µ B z t dµ dt y f, zf dµ + y i, z dµ, 3.6b where z is the solution of 3.6. The only difference between the cost functions defined by 3.5 for averaged controllability and 3.7 for simultaneous controllability is that, for simultaneous controllability, we allowed the final condition of the adjoint system to depend on the parameter. Assuming that J has a minimizer ẑ L, ; µ, we obtain, by computing the first variation of J, that: ût = B ẑt dµ t [, T ] a.e

10 It is clear that J is convex. Thus, proving the existence of a minimizer z f L, ; µ for J is equivalent to showing that J is coercive, i.e. to the existence of a constant ĉt > such that: ĉt z f dµ T B z t dµ where z is the solution of 3.6 with final condition z f. Summarizing this discussion, we end up with: dt z f L, ; µ. 3.9 Theorem 3.3. System. fulfills the exact simultaneous controllability property.8 if and only if the adjoint system 3.6 satisfies the exact simultaneous observability inequality 3.9. When these properties hold, the exact simultaneous control of minimal norm is given by 3.8, where ẑ is the solution of 3.6 with final condition ẑ f and ẑf L, ; µ is the minimizer of J defined by The case where exact simultaneous controllability fails Let us notice that very few systems have the property of simultaneous controllability. When card is finite, the situation is clear since simultaneous controllability follows from a Kalman rank on an augmented system see 4 th item of Remark.. But when card is infinite, the situation more complex. In propositions 3. and 3. we present two situations where simultaneous controllability cannot hold. Proposition 3.. Set, F, µ with = { j, j N} R where i j for every i j and F = P. Assume there exists an open interval I of R such that A, B admits an analytic extension on I and assume there exists I supp µ and a sequence jk k N I supp µ N such that lim k j k = and jk for every k N. Then the system. cannot be exactly simultaneously controllable. Example 3.. Consider the system ẏ = + y + u with { n, n N } = with a probability measure µ such that µ{} > for every. This system is not exactly simultaneously controllable although the truncated system in which we consider { n, n {,, N}} with the probability measure µ{} µ N given by µ N {} = { µ N, }, for { N,, n, n {,, N}} is simultaneously controllable, whatever N N is. In fact, for this truncated system, the precise values of the measure µ N are not important since its simultaneous controllability can be understood though the augmented system: d dt y. y N = N y. +. u. The Kalman matrix of this system is: N..., + N... + N N y N

11 which is a Vandermonde matrix of determinant i<j N i. j In addition, since this determinant goes to as N goes to, it is not surprising that the system with the full set of parameters { n, n N } is not simultaneously controllable. Proof of Proposition 3.. First of all, the solution of. with initial condition y i = and control u is: y t = t e t sa B us ds, t. Since A, B admits an analytic extension in I, y t can also be extend to an analytic function on I for every t >. Then, by analytic unique continuation, if for some K N we have y jk T = for every k > K, then y T = for every I. Thus, any final state such that y f jk = for k > K and y f for \ { j k, k > K} cannot be reached from the initial state y i =. Let us notice that a similar result holds when dealing with Lipschitz functions. Proposition 3.. Set, F, µ a probability space. Assume R d, with and F = B. Set and ε > such that B ε, with B ε the ball of R d centered on of radius ε. Assume µ Bε is a nonnegative regular measure and the map A, B is Lipschitz-continuous on B ε. Then the system. cannot be exactly simultaneously controllable. Example 3.. Let us consider the system ẏ = + y + u with [, ] = is the random parameter and for which we consider the probability measure µ given by dµ = d. Due to Proposition 3., this system is not simultaneously controllable since [, ] +, R R is Liepschitz-continuous and µ is a nonnegative regular measure. The proof of this result is based on the three following lemmas: Lemma 3.. Set, F, µ a probability space. Then the system ẏ = Ay + Bu cannot be exactly simultaneously controllable unless for every e F, µ e is either or. Remark 3.. Note that in this Lemma we are considering identical copies of the same system. The Lemma ensures that simultaneous controllability necessarily fails if we have more than one copy of the same system. Proof. Assume by contradiction there exists e F so that µ e, and the system ẏ = Ay +Bu is simultaneously controllable. Let us now consider the final target y f such that yf = for every e and y f for every \ e. Then it is clear that this final target cannot be reached from the initial point y i =. Lemma 3.. Set T >,, F, µ a probability space and let A, B be a bounded and integrable map and assume there exists ĉt > such that: T B eta z dµ dt ĉt z dµ z L, ; µ.

12 Let ε > and define an integrable map A ε, Bε such that: ess sup A A ε L + B B ε L, ε. Then, for small enough ε >, there exists a constant ĉ ε T > such that: T B ε e taε z dµ dt ĉ ε T z dµ z L, ; µ. This lemma ensures that, if system ẏ = A y +B u is exactly simultaneously controllable then, under small enough perturbation assumptions, the perturbed system A and B is still exactly simultaneously controllable. Proof. Since A ε, Bε is uniformly convergent to A, B as ε, we obtain: T B eta z B ε e taε z dµ with Cε > and lim ε Cε =. But, using Minkowski inequality, we obtain, for every ε >, T B ε e taε z dµ dt T T dt Cε z dµ z L, ; µ, B eta z dµ ĉt Cε This proves the desired result for small enough ε. dt B eta z B ε e taε z dµ dt z dµ z L, ; µ. Let us finally show that the exact simultaneous controllability property is inherited by reduction of the set of parameters. Lemma 3.3. Set T >,, F, µ a probability space and A, B L L,. Assume there exists ĉt > such that: T B eta z dµ Then for every ω F, such that µω, we have: T with µ ω = µ ω µω. ω B eta z dµ ω dt ĉt z dµ z L, ; µ. dt µωĉt z dµ ω z L ω, ; µ, ω

13 Proof. For every z L ω, ; µ, we define z L, ; µ by z = using the simultaneous controllability property 3.9, T ω B eta z dµ dt = T B eta z dµ { z if ω, if \ ω. We have, dt ĉt z dµ = ĉt z dµ. ω We are now in position to prove Proposition 3.. Proof of Proposition 3.. According to Lemma 3.3, in order to prove that the system is not exactly simultaneously controllable, it is enough to prove it on the reduced set of parameters ω = B ε with probability measure µ ω = µ ω µω by assumption, µω > and µω is nonnegative and regular. Let us assume by contradiction that the system. is exactly simultaneously controllable in a time T >. Hence, by duality, there exists a constant ĉ ω T > such that: T ω B eta z dµ ω dt ĉ ω T z dµ ω z L ω, ; µ ω. ω Furthermore, ω A, B is Lipschitz-continuous on the bounded domain ω of R d and thus it can be uniformly approximated by piecewise constant functions, i.e. for every ε >, there exists a piecewise constant map ω A ε, Bε such that: ess sup A A ε L + B B ε L, ε. ω Thus from Lemma 3., for ε > small enough, there exists ĉ ω,ε T > so that: T ω B ε e taε z dµ ω dt ĉ ω,ε T z dµ ω z L, ; µ ω. ω That is to say that for ε > small enough, the system ẏ = A ε y + B ε u is exactly simultaneously controllable. But, since µ ω is nonnegative and regular, there exists a domain D ε ω with nonempty interior such that µ ω D ε > and A ε, Bε is constant on D ε. sing again Lemma 3.3, the system ẏ = A ε y +B εu has to be exactly simultaneously controllable for the probability space D ε, BD ε, µω Dε µ ω Dε µ ω D ε µ ω D ε. Notice that is a regular measure. Since A ε, Bε is constant on D ε, from Lemma 3. we have µ ω e is either µ ω D ε or, for every e BD ε. This contradicts µω Dε µ ω D ε regular. Propositions 3. and 3. told us that it is hard to build continuously-dependent parameter systems which are exactly simultaneously controllable. However, as we have seen, the averaged controllability property holds for a variety of models. Consequently, it is natural to look for averaged controls which are optimal in the sense that they minimize the output s variance. This is the core of section 4. 3

14 3.4 Momentum approach for simultaneous controllability In 3., we gave a necessary and sufficient condition, 3.9, for simultaneous controllability. However, even on simple problems, it is difficult to check whether this condition is satisfied or not. In this paragraph, we present an iterative approach to check whether the observability inequality 3.9 is fulfilled or not. The method presented here can also be seen as an alternative method to the one we proposed in the rest of this paper see section 4 in order to link averaged controllability to exact simultaneous controllability. To simplify the notation we define the operator E LL, ; µ, by: Ey = y dµ y L, ; µ. 3. Notice that we have E z = z and E E = Id. Let us first remind that proving the averaged controllability property is equivalent as proving that the cost function J defined by 3.5 is coercive and proving the exact simultaneous controllability is equivalent as proving that the cost function J defined by 3.7 is coercive. In addition, we have also noticed that we have J = J E, where E is given by 3.. Thus, proving that J is coercive means proving that the restriction of J to the subset E = { y, y } of L, ; µ is coercive. Let us also notice that since L, R; µ is an Hilbert space, one can define an orthonormal basis ϕ i i I with the convention I and ϕ = of this space. Based on the above construction of E, we define for every i I, the operator E i LL, ; µ, by: E i y = y ϕ i dµ y L, ; µ, 3. so that L, ; µ = i I E i. Let us assume that L, R; µ is a separable Hilbert space, that is to say that we can choose I = N if L, R; µ is of infinite dimension or I = {,, d} N if L, R; µ is of dimension d. For every k N, we define the finite dimensional subspace V k of L, ; µ by: V k = k E i L, ; µ. 3. i I i k Let us also define the constant ĉ k T by: ĉ k T = inf z f V k \{} T B z t dµ z f L,:µ dt k N, 3.3 that is to say: ĉ k T z f dµ T B z t dµ dt k N z f V k, with z t the solution of the adjoint problem 3.6 with final condition z T = z f V k. Thus, if ĉ k T >, J is convex and coercive on V k. 4

15 Since V k V k+, it remains clear that the sequence ĉ k T k N is decreasing. In addition, one can easily convince that if lim ĉkt >, there exists ĉt > ĉt = lim ĉkt such that: k k ĉt z f dµ T B z t dµ dt z f L, ; µ. That is to say that J is convex and coercive on L, ; µ and hence we have exact simultaneous controllability. Moreover, the sequence z f k, V k of minimizer of J on the finite dimensional subspace V k of L, ; µ is convergent to a minimizer ẑ L, ; µ of J on L, ; µ. Summarizing the above discussion, leads to the following: Theorem 3.4. Assume that L, R; µ is a separable Hilbert space and let ϕ i i I with {} I N and the convention ϕ = be an orthonormal basis of this space. For every j I, we define the map E j LL, ; µ, by 3. and, for every k N, we define the finite dimensional subspace V k of L, ; µ by 3. and J k the restriction of J defined by 3.7 on V k. Let us also introduce for every k N, the constant ĉ k T defined by 3.3. Then, the system. is exact simultaneously controllable if and only if inf k I ĉkt >,. Moreover, if this property holds, for every k N, J k admits a minimum ẑ f k, V k and as k goes to, ẑ f k, is convergent in L, ; µ to an element ẑ f which minimize J and the exact simultaneous control of minimal L -norm is given by: ut = B ẑt dµ t [, T ] a.e., where ẑ is the solution of 3.6 with final condition ẑ f. Remark 3... It is obvious that if there exists k N such that ĉ k T = the system. cannot be exactly simultaneously controllable.. The case ĉ k T > for every k N and lim k ĉkt = is undetermined. However, for this case, we might recover the property of approximate simultaneous controllability. 3. Let us mention that the property: ĉ k T > s.t. z f V k, ĉ k T correspond to a Kalman rank condition. z f dµ T B z t dµ dt, 3.4 More precisely, k z f V k means there exists z j,f j=,,k k+ such that z f = ϕ j z j,f. Let us then denote by z j the solution of 3. with final condition zj,f. Due to linearity, the solution z of 3.6 with final condition z f is: z t = k ϕ j z j t = j= 5 k ϕ j e T ta z j,f. j= j=

16 Finally, since we are in a finite dimensional space the coercive property 3.4 is equivalent to the uniqueness property: B j= k ϕ j e T ta z j,f dµ = t [, T ] a.e. We conclude by time analyticity that 3.4 holds if and only if: [ ] rank  l ˆB dµ, l N = k + dim, where we have defined: A  =... A L k+ and ˆB = = z j,f = j {,, k}. ϕ B. ϕ k B L, k+. Even if 3.4 can be easily obtained by the use of Kalman rank condition, in order to conclude that the system satisfies the exact simultaneous controllability, we need an estimate on the constants ĉ k T which is hard to obtain. This is why in section 4 we present another approach based on a penalty problem. 4. When Card <, the moments are solution of an ordinary differential equation. More precisely, consider = {,, K} with measure µ given by µ{k} = θ k with θ k, and K k= θ k =. Let us consider an orthonormal basis {ϕ,, ϕ K } of L, R; µ with the convention, ϕ k =. Then the i th -momentum is: with: y = y. y K Thus, setting: M = Y i = K θ k ϕ i ky k = M i I y i {,, K }, k= K, M i = M. M K ϕ i θ Id ϕ i K θ K Id L K, A L K, A =... A K and I = θ Id L K... and B = θk Id B. B K L K. L, K, 6

17 the momentums Y = Y. Y K satisfies noticing that MM = Id K : Ẏ = MIAI M Y + MIBu. 3.5 Controlling the first k momentums of y k k means controlling the first k dim components of Y, solution of 3.5. Since the basis ϕ, ϕ,, ϕ K is free except ϕ = one can consider the problem of finding the best possible basis. For instance we can wonder if there exists ϕ,, ϕ K such that the pair MIAI M, MIB has a normal form see [, Proposition..6]. That is to say find ϕ,, ϕ K such that MIAI M has the structure and MIB the structure. 4 A penalty method linking averaged and simultaneous controllability As in all this paper, we assume in this section that the assumptions of lemmas. and. are satisfied. In this section, we will present our strategy to link averaged controllability to exact simultaneous controllability. First of all, solving the averaged control problem, can be done with the Hilbert niqueness Method, that is to say minimize the L -norm of the control with the constraint y T dµ = y f dµ. Thus, using Euler-Lagrange formulation or directly Theorem 3., on can see that the averaged control of minimal L -norm is given by 3.4. In order to reduce the output s variance, one can think to penalise the cost function J given by J u = u L [,T ], with the output s variance, y T y f dµ. Thus, we introduce the penalty problem of optimisation: min J κ u := ut L [,T ], + κ y T ; u y f L,;µ E y T ; u y f κ, 4. =. where in the above, y is the solution of. defined by. with control u, L, ; µ is the Hilbert space introduced in.3 and E is the expectation defined by 3.. Let us give an existence result. Proposition 4.. If system. satisfies the averaged controllability property.4 then for every T >, y i, y f L, ; µ and κ, the minimisation problem 4. admits one and only one solution u κ L [, T ],. In addition, the optimal control u κ satisfies: u κ t = B z t dµ t [, T ], 4.a where, z is solution of: with z unknown. ż = A z, z T = z + y f y T ; u κ, 4.b 7

18 Proof. For every κ, it is clear that J κ is convex. Since we have assumed that the system. satisfies the averaged controllability property.4, this ensure that the set: { u L [, T ],, E y T ; u y f } = is non empty and in addition, this set is a convex and closed set of L [, T ],. Moreover, the averaged controllability property ensure that J is coercive on this set and consequently J κ is also coercive on this set. Thus, there exists a unique minimizer u κ L [, T ], for the minimisation problem 4.. Let us now prove the optimality conditions. Let us define the Lagrangian of the system: The optimality conditions are: Lu, z = J κ u + z, E y T ; u y f u L [, T ],, z. But we have, u Lu, z = u + control u κ should satisfy 4.. z L = and u L =. B et ta z + y T ; u y f dµ. That is to say that, the optimal Of course, we have introduced the cost functions J κ in order to pass to the limit κ. Let us first state a trivial statement: Lemma 4.. Set T > and assume the system. is controllable in average. For every κ, let us define u κ the minimum of J κ under the constraint E y T, u κ y f =. Then, we have: u κ L [,T ], u κ+ε L [,T ], and y T ; u κ y f L,;µ y T ; u κ+ε y f L,;µ κ, ε. In addition, for every κ, we have: y T, u κ y f L,;µ { y = min T ; u y f L,;µ, u L [, T ],, u L [,T ], u κ L [,T ], and E y T ; u y f } =. 4.3 Proof. It remains clear that for every κ, ε, we have: J κ u κ J κ u κ+ε J κ+ε u κ+ε J κ+ε u κ. Thus from, J κ u κ +J κ+ε u κ+ε J κ u κ+ε +J κ+ε u κ, it is easy to see that y T ; u κ y f L,;µ is decreasing and then, form J κ u κ J κ u κ+ε, we obtain that u κ L [,T ], is increasing. κ Let us now prove 4.3. To this end, we assume by contradiction that there exists u L [, T ], such that: u L [,T ], u κ L [,T ],, E y T ; u y f = and y T ; u y f L,;µ < y T ; u κ y f L,;µ. Then we have J κ u < J κ u κ which is in contradiction with u κ minimize J κ. κ 8

19 Various situations could hold as κ. These different situations, reported on Table, are given by the following theorem. Theorem 4.. Set T > and assume that the system. in controllable in average in time T. For every κ, let us define u κ the minimum of J κ under the constraint E y T, u κ y f =. Define y L, ; µ as the minimizer of: min y y f L,;µ y {y T ; u, u L [, T ], }, Ey = Ey f. 4.4 Then, the following alternative holds: If u κ L [,T ], κ is bounded, then up to a subsequence, u κ κ converges to a control which steers exactly y i to y and realises the minimum of: min u L [,T ], y T ; u y L,;µ =. If u κ L [,T ], κ is unbounded, then yi can be approximatively steered to y. y In addition, if lim T ; u κ y f κ L =, then we have y,;µ = yf. Proof. Without loss of generality, we can assume that y i =. Let us first notice that y L, ; µ is well defined. In fact, y is the orthogonal projection of y f in L, ; µ on the closed vector space {y T ; u, u L [, T ], } { y, Ey = Ey f }. Let us assume u κ L [,T ], κ bounded. From Lemma 4., the sequence u κ L [,T ], κ is increasing, hence there exists u L [, T ], such that up to a subsequence, u κ κ is weakly convergent to u and in addition, we have: u L [,T ], lim κ u κ L [,T ],. Since u κ κ is weakly convergent to u, it is easy to obtain that y T ; u κ κ is weakly convergent to y T ; u L, ; µ. Hence, E y T ; u = E y f and y T ; u y f In addition, from Lemma 4., the sequence have: L,;µ y T ; u κ y f L,;µ lim infy T ; u κ y f κ L,;µ. is decreasing thus, we y T ; u y f L,;µ y T ; u κ y f L κ >,;µ and hence, from relation 4.3 of Lemma 4., we obtain u L [,T ], u κ L [,T ], that is to say, u L [,T ], = lim u κ L κ [,T ], and up to a subsequence, u κ κ is strongly convergent κ 9

20 to u in L [, T ],. Consequently, y T ; u κ κ is strongly convergent to y T ; u in L, ; µ. Let us now prove that y T, u = y. Assume by contradiction that it is not the case. That is to say there exists ū L [, T ], such that: E y T ; ū = E y f and y T ; ū y f L,;µ < y T, u y f L,;µ. On the other hand, we have J κ u κ J κ ū for every κ, i.e.: ū L κ [,T ], u κ L [,T ], y T ; u κ y f L,;µ y T ; ū y f κ >. L,;µ Taking the limit κ comes the contradiction: y T ; ū y f L,;µ y T ; u y f L,;µ. Finally, it remains clear that lim y T ; u κ y f κ L,;µ = is equivalent as y = yf. Let us assume that u κ L [,T ], is not bounded. κ The results of this point are direct consequences of 4.3 given in Lemma 4.. If the system. is simultaneously controllable, then we do not need the extraction of a subsequence procedure and the convergence rates to the simultaneous control and the variance are linked. Proposition 4.. Assume system. is exactly simultaneously controllable in time T >. Let u L [, T ], be the exact simultaneous control of minimal norm steering y i to yf and let u κ L [, T ], be the minimizer of 4.. Then, u κ κ is strongly convergent to u and, in addition, y T, u κ y f L,;µ u L [,T ], κ u κ u L [,T ],. 4.5 Proof. Let us first prove that the sequence u κ κ is strongly convergent to u. First of all, since we assumed that the system. is exactly simultaneously controllable, the minimisation problem: min u L [,T ], y T ; u y f L,;µ =, admits one and only one minimizer u L [, T ],. Regarding to the proof of Theorem 4., there exist a nondecreasing sequence κ n n N such that u κn n N is strongly convergent to u. Let us assume by contradiction that there exists another sequence κ n n N such that u κn n N is not convergent to u. But from Lemma 4., we have u κn L [,T ], u L [,T ], and hence, u κn n N is weakly convergent to some ũ L [, T ],, with ũ u. As in the proof of Theorem 4., we can prove that this convergence is strong. Consequently, ũ L [,T ], = u L [,T ], and since ũ u, y T ; ũ y f L,;µ >. But since y T ; u κ y f L,;µ κ is decreasing, for every n N, there exists n N such that: y T ; ũ y f L,:µ y T ; u κn y f L,:µ y T, u κn y f L,;µ.

21 But when n, we n can also be choosen so that it goes to infinity. Thus, y T ; ũ y f L,:µ = y T ; u y f L,:µ = leading to a contradiction. Let us now prove 4.5. First of all, changing y f in yf e T A y i, we can assume without loss of generality that y i =. Set u κ = u + v κ, then v κ is a minimizer of: We have: min Thus, for every κ, κ T G κ v = v L [,T ], + v, u L [,T ], + κ T E e T ta B vt dt =. e T ta B v κ t dt G κ v κ G κ =. dµ v κ L [,T ], + κ T T e T ta B vt dt e T ta B v κ t dt dµ dµ v κ, u L [,T ], v κ L [,T ], u L [,T ],. Let us now give the consequences of Theorem 4. in the case where the cardinal of is finite. Corollary 4.. Assume L, ; µ is of finite dimension. Then the sequence of minimizers û κ κ of the optimisation problem 4. is strongly convergent up to the extraction of a subsequence to an element û L [, T ], satisfying the minimisation problem: min u L [,T ], y T = y µ a.e., where y is defined by Theorem 4., i.e. is the minimizer of 4.4. A graphical interpretation of this result is given on Figure. Proof. Let us use the notations introduced in Theorem 4.. Since L, ; µ is a finite dimensional space, { y T ; u, u L [, T ], } { y, Ey = Ey f } is a closed affine subspace of L, ; µ. Consequently, there exists u L [, T ], such that y = y T ; u. Example 4.. This example illustrates the result of Corollary 4. in the exact simultaneous controllability case. Consider the probability space = {, } and the probability measure µ is given by µ{} = µ{} =. The parameter dependent system under consideration is: ẏ = Ay + Bu y = y i, with A =, B = and y i =. We fix the final target to y f = and the final time T to be.

22 { } Ey = Ey f y f {yt ; u, u L [, T ], } y Figure : nder the assumptions of Corollary 4., at the limit κ, the emergent control will be a control steering y i to y. The corresponding augmented system is: ẏ = Ay + Bu y = y i, A with A = = B A, B = = y B and i yi = y i =. sing the Kalman rank condition, it is easy to see that this system is controllable in the classical sense and controlling the average means controlling y + y 3, y + y 4. On figures 3, 4 and 5, we plot the numerical results dealing with the averaged control, the exact simultaneous control and the solution of the penalisation problem, when letting the parameter κ growing.. Trajectories for average control 6 Average control control average = a Controlled trajectories in the phase plan using the averaged control. The variance at final time is.75e t b Averaged control, the norm of the control is 3.9. Figure 3: On left, we plotted the trajectories obtained by the averaged control right which is of minimal L -norm.

23 3 Trajectories for simultaneous control average = 5 Simultaneous control control a Controlled trajectories in the phase plan using the simultaneous control t b Simultaneous control, the norm of the control is 6.34e+. Figure 4: On left, we plotted the trajectories obtained by the simultaneous control right which is of minimal L -norm. Control s convergence with respect to κ uκ u L,T ; Variance with respect to κ variance.... e+6 κ a Plot of the L -distance between the exact simultaneous control and the optimal control of the minimisation problem indexed with κ. This distance behaves as Cκ α with α.98. e 5 e+6 κ b Plot of the variance at final state y T y f dµ with respect to κ. The variance behaves as Cκ α with α.95. Figure 5: Plots in log log scale of the L -distance between the solution of the optimal control with parameter κ and the exact simultaneous control left and of the variance at final state right as κ grows. The decay rates obtained are coherent with the results of Proposition 4.. Example 4.. This example illustrate the result of Corollary 4. when the is no simultaneous controllability. For this example, we consider again the probability space = {, } and the probability density µ given by µ{} = µ{} =. The parameter dependent system under consideration is: y = A y + Bu y = y i, if =, with B =, y i = and A = if =. sing the Kalman rank condition, introduced by E. Zuazua see Theorem 3., one can see that this system is controllable in average. On the other hand, the simultaneous controllability of this system reduce to 3

24 prove the classical controllability of the augmented system: ẏ = Ay + Bu, A with A = = B A and B = = B. One can easily see that rank [ B, AB, A B, A 3 B ] = 3 < 4 and hence, the Kalman rank condition is not satisfied. On figures 6, 7 and 8, we present the numerical results for this system. As in Example 4., the final time T is set to and the target y f is. 3 Trajectories for average control average = Average control control a Controlled trajectories in the phase plan using the averaged control. The variance at final time is.3e t b Averaged control, the L -norm of the control is.99e+. Figure 6: On left, we plotted the trajectories obtained by the averaged control right which is of minimal L -norm L -norm of the control with respect to κ Variance with respect to κ variance norm of the control κ a Plot of the norm of the control with respect to κ κ b Plot of the variance at final state with respect to κ. Figure 7: Plots of the norm of the control left and of the variance at final state right as κ grows. 4

25 4 3 Trajectories for min-variance control average = a Controlled trajectories in the phase plan using the optimal control for κ = The variance at final time is min-variance control control t b Optimal control for κ = 5. 3 its L -norm is 5.3e+. Figure 8: On left, we plotted the trajectories obtained by the optimal control right for κ = Numerical realisation when Card is infinite In this section we will study the discrete event case = N. For this case, we consider the probability space N, PN, µ. A natural way to deal with this problem is to truncate it. More precisely, instead of considering the probability space N, PN, µ, we consider the probability space N, PN, χ Z µ with the measure χ Z µ given by µ{} if Z, χ Z µ{} = µ{,, Z} Z N, N, 5. otherwise, for Z N large enough so that µ{,, Z} >. Since our penalisation procedure needs the system ẏ = A y + B u to be controllable in average the first question we should answer is whether this averaged controllability property is stable or not through the truncation procedure. Proposition 5.. Assume the system. is controllable in average for the measure µ. Then there exists Z N such that for every Z Z, this system is controllable in average for the measure χ Z µ given by 5.. Let us also notice that this truncation procedure does not affect the simultaneous controllability property for Z large enough. More precisely, by direct application of Lemma 3.3, we have: Proposition 5.. Assume the system. is exactly or approximatively simultaneously controllable for the measure µ. Then for every Z N such that µ {,, Z} >, this system is simultaneously controllable for the measure χ Z µ given by 5.. Remark 5.. Notice that by truncation, one can lose the averaged controllability property. This is for instance the case of the system considered in Example 4.. In opposition the simultaneous controllability property cannot be lost by truncation. This is natural since if the system in simultaneously controllable, all the events y,, y Z can be exactly controlled. Consequently, if a system is simultaneously controllable, then it is controllable in average and each of its truncation is controllable in average. 5

26 Proof of Proposition 5.. Set θ = µ{} without loss of generality, we can assume that θ > for every θ N. Set θ Z = χ Z Zµ{} = θ if Z, otherwise. Let us remind that due to Theorem 3., the pairs A, B being controllable in average, is equivalent as 3.3: T c z f z f θ dt z f. with c = ct > independent of z f. But, T Z B eta z f θ Z and using Minkowsky inequality, N B eta dt = Z θ T Z B eta z f θ dt Z θ T Z B eta z f θ Z T dt B eta z f θ dt From the averaged controllability property, there exists c > such that: c z f T B eta z f θ and due to the admissibility condition, there exists C > such that: T =Z+ B eta z f θ dt C z f dt T =Z+ =Z+ Consequently, T Z B eta z f θ Z c C Z θ dt Z θ z f. c C Z θ Since lim Z Z θ = c >, we obtain the result. θ. B eta z f θ dt. 6

27 Let us finally study the error between the initial minimisation problem: min J κ u := T ut dt + κ y T ; u y f µ{} = y T ; u y f µ{} κ 5. and the truncated minimisation problem: min J Z κ u := Z T ut dt + κ Z y T ; u y f χ Zµ{} y T ; u y f χ Z µ{} = κ, Z Z, 5.3 with Z N given by Proposition 5.. Proposition 5.3. Assume that the system. is controllable in average for the probability measure µ. Set κ. Let u Z κ be a minimizer of the truncated minimisation problem 5.3. Then, as Z, the sequence u Z κ Z is strongly convergent in L [, T ], to the minimizer u κ of the initial minimisation problem 5.. Proof. Without loss of generality, we can assume that µ{} > for every N and for convenience, θ we set µ{} = θ = θ and as previously, θ Z = χ Z Zµ{} = θ if Z, otherwise. Without loss of generality, we can also assume that for every Z N, the system. in controllable in average for the probability measure χ Z µ. Let us introduce for every Z N {} the map I Z : L [, T ], {, } defined by: Z I Z if y T ; u y f θ Z u = =, otherwise. Thus minimizing Jκ Z under the constraint Z y T ; u y f θ Z = is equivalent as minimizing Jκ Z + I Z. The proof of this result is based on Γ-convergence. More precisely, we will prove that the sequence J Z κ + I Z Γ-converge to J Z N κ + I. pper bound: Let u Z Z N L [, T ], N be strongly convergent to an element u L [, T ],. The aim of this point is to prove: J κ u + I u lim inf Z J Z κ u Z + I Z u Z

28 If lim inf Z IZ u Z =, then, it is clear that 5.4 is true. Otherwise, we can assume up to the extraction of a subsequence that for every Z N, we have I Z u Z =. nder this assumption, let us prove: I u = and lim Z Z y T ; u Z y f θz = y T ; u y f θ. This ensure Let us prove that I u = : To this end, let us notice: y T, u y f θ = = = y T, u y f θ y T, u y f θ + µ{,, Z} y T, u Z y f θ Z µ{,, Z} =Z+ T e T ta B u t + µ {,, Z} + + µ{,, Z} µ{,, Z} =Z+ =Z+ y T, u Z y f θ u Z t µ {,, Z} T y T, u Z y f θ dt θ e T A y i yf θ e T ta B u Z t dt θ e T A y i yf θ The admissibility condition, ensure: T e T ta u Z t B u t dt θ µ {,, Z} C u u Z µ {,, Z} L [,T ],, with C > a constant. 8

29 sing Cauchy-Schwarz inequality, we obtain: =Z+ T e T ta B u Z t dt θ µ{,, Z} µ{,, Z} T =Z+ T e T ta B u Z t dt e T ta B u Z t dt θ θ But, according to the admissibility conditions see Lemma., there exists a constant Ĉ > such that: T e T ta B u Z t dt θ µ{,, Z} Ĉ u Z L [,T ],. =Z+ Thus, taking the limit Z, we obtain y T, u y f θ. Let us prove lim Z Z y T ; u Z y f θz = y T ; u y f θ : For every Z N, we have, by Cauchy-Schwarz inequality: =, i.e. I u =. Z y T ; u Z y f θz Z = y T ; u Z y T ; u Z θz + y T ; u y f + θz Z y T ; u Z y T ; u, y T ; u y f Z y T ; u Z y T ; u θz Z + y T ; u y f θz. θz sing the admissibility of every system indexed by, for every N, there exists C > such that: Z y T ; u Z y T ; u θz Z C θ Z u Z u L [,T ],. 9

30 In addition, due to assumption.3 made in Lemma., we have hence, since u Z Z is strongly convergent to u, lim Z Z C θ Z < and lim Z Z y T ; u Z y T ; u θz =. On the other hand, it remains clear, due to the construction of θ Z that: Thus, lim Z lim Z Z y T ; u y f Z y T ; u Z y f θz θz = y T ; u y f θ. = y T ; u y f θ. strongly conver- Lower bound: Set u L [, T ],. The aim is to prove that there exists a sequence u Z Z N gent to u such that: J κ u + I u lim sup J Z κ u + I Z u. Z If I u = then this result can be easily obtained with u Z = u. Let us now assume that I u =. From the previous point, it remains clear that if the sequence u Z Z is converging to u and if for every Z N, I Z u Z = then: J u = lim Z J Z κ u Z. Thus we only need to prove that such a sequence u Z Z exists. Let us write u Z = u + v Z. Then I Z u Z = means: Z T e T ta B v Z t dt θ Z = Z y T ; u y f θ Z. Since we assumed that the system. is controllable in average, such a v Z exists and in addition, there exists a constant C > independent of v Z such that: v Z L [,T ], C Z y T ; u y f θ Z. 3