Approximate and Approximate Null-Controllability of a Class of Piecewise Linear Markov Switch Systems

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Approximate and Approximate Null-Controllability of a Class of Piecewise Linear Markov Switch Systems

Dan Goreac, Claudia Grosu, Eduard Rotenstein

To cite this version:

Dan Goreac, Claudia Grosu, Eduard Rotenstein. Approximate and Approximate Null-Controllability

of a Class of Piecewise Linear Markov Switch Systems. Systems and Control Letters, Elsevier, 2016,

96, pp.118-123. �10.1016/j.sysconle.2016.07.003�. �hal-01266268v2�

Approximate and Approximate Null-Controllability of a Class of Piecewise Linear Markov Switch Systems ^y

Dan Goreac

Alexandra Claudia Grosu

Eduard-Paul Rotenstein

June 27, 2016

Abstract

We propose an explicit, easily-computable algebraic criterion for approximate null-controllability of a class of general piecewise linear switch systems with multiplicative noise. This gives an answer to the general problem left open in [13]. The proof relies on recent results in [4] allowing to reduce the dual stochastic backward system to a family of ordinary di¤erential equations. Second, we prove by examples that the notion of approximate controllability is strictly stronger than approximate null-controllability. A su¢ cient criterion for this stronger notion is also provided. The results are illustrated on a model derived from repressed bacterium operon (given in [19] and reduced in [5]).

1 Introduction

This short paper aims at giving an answer to an approximate (null-)controllability problem left open in [13]. We deal with Markovian systems of switch type consisting of a couple mode/ trajectory denoted by ( ; X): The mode component evolves as a pure jump Markov process and cannot be controlled. It corresponds to spikes inducing regime switching. The second component X obeys a controlled linear stochastic di¤erential equation (SDE) with respect to the compensated random measure associated to . The linear coe¢ cients governing the dynamics depend on the current mode.

The controllability problem deals with criteria allowing one to drive theXT component arbitrarily close to ac- ceptable targets. An extensive literature on controllability is available in di¤erent frameworks: …nite-dimensional deterministic setting (Kalman’s condition, Hautus test [14]), in…nite dimensional settings (via invariance criteria in [22], [6], [21], [17], [16], etc.), Brownian-driven control systems (exact terminal-controllability in [20], approx- imate controllability in [3], [9], mean-…eld Brownian-driven systems in [12], in…nite-dimensional setting in [8], [23], [1], [10], etc.), jump systems ([11], [13], etc.). We refer to [13] for more details on the literature as well as applications one can address using switch models.

The paper [13] provides some necessary and some su¢ cient conditions under which approximate controllability towards null target can be achieved. In all generality, the conditions are either too strong (su¢ cient) or too weak (only necessary). Equivalence is obtained in [13] for particular cases : (i) Poisson-driven systems with mode-independent coe¢ cients and (ii) continuous switching. In the present paper, we extend the work of [13]

and give explicit equivalence criterion for the general switching case. The approach relies, in a …rst step, as it has already been the case in [13, Theorem 1], on duality techniques (brie‡y presented in Subsection 2.1).

However, the intuition on this new criterion and its proof are extensively based on the recent ideas in [4]. The dual backward stochastic system associated to controllability is interpreted as a system of (backward) ordinary di¤erential equations in Proposition 12. Reasoning on this new system provides the necessary and su¢ cient criterion for approximate null-controllability for general switching systems with mode-dependent multiplicative noise (Theorem 6 whose proof relies on Propositions 13 and 14). As a by-product, we considerably simplify the proofs of [13, Criteria 3 and 4] (in Subsection 2.3). Second, we give some elements on the stronger notion of

AMS MSC:93B05, 93B25, 60J75,Keywords: Approximate (null-)controllability Controlled Markov switch process Invari- ance Stochastic gene networks

yAcknowledgement. The work of the …rst author has been partially supported by he French National Research Agency project PIECE, number ANR-12-JS01-0006. The work of the third author has been partially supported by the Grant PN-II-ID-PCE- 2011-3-0843, no. 241/05.10.2011, Deterministic and stochastic systems with state constraints.

zUniversité Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, F-77454, Marne-la-Vallée, France, [email protected]

xFaculty of Mathematics, “Alexandru Ioan Cuza” University, Bd. Carol I, no. 9-11, Iasi, Romania

{Faculty of Mathematics, “Alexandru Ioan Cuza” University, Bd. Carol I, no. 9-11, Iasi, Romania

(general) approximate controllability. While the notions of approximate and approximate null-controllability are known to coincide for Poisson-driven systems with mode-independent coe¢ cients, we give an example (Example 9) showing that this is no longer the case for general switching systems. Furthermore, we show that the condition exhibited in [13, Proposition 3] in connection to approximate null-controllability is actually su¢ cient for general approximate controllability (see Condition 10). The proof follows, once again, from the deterministic reduction inspired by [4]. The theoretical results are illustrated on a model derived from repressed bacterium operon (given in [19] and reduced in [5]).

We begin with presenting the problem, the standing assumptions and the main results: the duality abstract characterization in Theorem 2, the explicit criterion in Theorem 6. We give a considerably simpli…ed proof of the results in [13] in Subsection 2.3. We discuss the di¤erence between null and full approximate controllability in Subsection 2.4, Example 9 and give a su¢ cient criterion for the stronger notion of approximate controllability (Criterion 10). Section 3 focuses on an example derived from [19] (see also [5]). The proofs of the results and the technical constructions allowing to prove Theorem 6 are gathered in Section 4.

2 The Control System and Main Results

We brie‡y recall the construction of a particular class of pure jump, non explosive processes on a space and taking their values in a metric space (E;B(E)): Here, B(E) denotes the Borel -…eld of E: The elements of the space E are referred to as modes. These elements can be found in [7] in the particular case of piecewise deterministic Markov processes (see also [2]). To simplify the arguments, we assume thatE is …nite and we let p 1be its cardinal. The process is completely described by a couple( ; Q);where :E !R+and the measure Q:E ! P(E), whereP(E)stands for the set of probability measures on(E;B(E))such that Q( ;f g) = 0:

Given an initial mode 02E;the …rst jump time satis…esP^{0; 0}(T1 t) = exp ( t ( 0)):The process t:= 0; on t < T 1: The post-jump location ¹ hasQ( 0; )as conditional distribution. Next, we select the inter-jump time T2 T1 such that P^{0; 0} T2 T1 t = T1; ¹ = exp t ¹ and set t:= ¹;ift2[T1; T2):The post- jump location ²satis…esP^{0; 0 2}2A = T2; T1; ¹ =Q ¹; A ;for all Borel setA E:And so on. To simplify arguments on the equivalent ordinary di¤erential system, following [4, Assumption (2.17)], we will assume that the system stops after a non-random, …xed number M >0 of jumps i.e. P^{0; 0}(TM+1=1) = 1. The reader is invited to note (see Remark 5) that, for large M; the criteria given in the main result (Theorem 6) no longer depend onM (due to the …nite dimension of the mode and state spaces).

We look at the process under P^{0; 0} and denote by F⁰ the …ltration F^[0;t] := f ^r:r2[0; t]g _{t 0}: The predictable -algebra will be denoted by P⁰ and the progressive -algebra by P rog⁰: As usual, we introduce the random measure q on (0;1) E by setting q(!; A) = P

k 11(^Tk(!); _Tk(!)(!))2A; for all ! 2 ; A 2 B(0;1) B(E): The compensated martingale measure is denoted by q. (For our readers familiar with [13],e we emphasize that the notation is slightly di¤erent, the counting measure qcorresponds to pin the cited paper and the martingale measure eqreplacesqin the same reference. Further details on the compensator are given in Subsection 4.1.)

We consider a switch system given by a process (X(t); (t)) on the state space R^N E; for someN 1 and the family of modes E. The control state space is assumed to be some Euclidian space R^d; d 1. The component X(t)follows a controlled di¤erential system depending on the hidden variable . We will deal with the following model (Ais implicitly assumed to be0 after the last jump).

(1) dX_s^x;u= [A( _s)X_s^x;u+Bu_s]ds+ Z

E

C( _s ; )X_s^x;uqe(ds; d ); s 0; X₀^x;u=x:

The operatorsA( )2R^{N N} ,B 2R^{N d}andC( ; )2R^{N N}, for all ; 2E. For linear operators, we denote by kertheir kernel and byImthe image (or range) spaces. Moreover, the control process u: R+ !R^d is an R^d-valued, F⁰ progressively measurable, locally square integrable process. The space of all such processes will be denoted by U^ad and referred to as the family of admissible control processes. The explicit structure of such processes can be found in [18, Proposition 4.2.1], for instance. Since the control process does not (directly) intervene in the noise term, the solution of the above system can be explicitly computed with Uad processes instead of the (more usual) predictable processes.

2.1 The Duality Abstract Characterization of Approximate Null-Controllability

We begin with recalling the following approximate controllability concepts.

De…nition 1 The system (1) is said to be approximately controllable in timeT >0starting from the initial mode

0 2 E; if, for every F[0;T]-measurable, square integrable 2 L² ;F[0;T];P^{0; 0};R^N , every initial condition x2R^N and every " >0, there exists some admissible control processu2 U^ad such that E^{0; 0}

h

jX_T^x;u j² i

":

The system (1) is said to be approximately null-controllable in time T > 0 if the previous condition holds for

= 0(P^0;0-a.s.).

At this point, let us consider the backward (linear) stochastic di¤erential equation (2)

(

dY_t^{T ;} = h

A ( t)Y_t^{T ;} R

E(C ( t; ) +I)Z_t^{T ;} ( ) ( t)Q( t; d ) i

dt+R

EZ_t^{T ;} ( )q(dt; d ); Y_T^{T ;} = 2L² ;F[0;T];P^{0; 0};R^N :

Classical arguments on the controllability operators and the duality between the concepts of controllability and observability lead to the following characterization (cf. [13, Theorem 1]).

Theorem 2 ([13, Theorem 1]) The necessary and su¢ cient condition for approximate null-controllability (resp.

approximate controllability) of (1) with initial mode 02E is that any solution Y_t^{T ;} ; Z_t^{T ;} ( ) of the dual sys- tem (2) for which Y_t^{T ;} 2kerB ; P^{0; 0} Leb almost everywhere on [0; T] should equally satisfy Y₀^{T ;} = 0;

P^0;0 almost surely (resp. Y_t^{T ;} = 0;P^{0; 0} Leb a:s:).

Remark 3 Concerning the operatorA; it is assumed to be a switched matrix but it could also depend on(t; t) or on all the times and marks prior to t: This is why, we implicitly assumed that A = 0 after the last jump (M^th) occurs. Similar assertions are true forC (otherwise, the backward equation (2) should be written with the compensator bqreplacing ( t)Q( t; d ).) The reader may also look at the end of Subsection 4.1.

2.2 Main Result : An Iterative Invariance Criterion

Before stating the main result of our paper, we need the following invariance concepts (cf. [6], [22]).

De…nition 4 We consider a linear operatorA 2R^{N N} and a family C= (Ci)_{1 i k} R^{N N}. (i) A setV R^N is said to be A- invariant ifAV V:

(ii) A setV R^N is said to be(A;C)- invariant ifAV V + Pk i=1

ImCⁱ:

We construct a mode-indexed family of linear subspaces ofR^N denoted by V^M;n

0 n M; 2E by setting

(3) A ( ) :=A ( )

Z

E

(C ( ; ) +I) ( )Q( ; d )andV^M;M = kerB ; for all 2E;and computing, for every0 n M 1;

(4) V^M;n the largest A ( ) ; h

(C ( ; ) +I) _V^M;n+1: 2E; Q( ; )>0 i

invariant subspace of kerB : Here, _V denotes the orthogonal projection operator onto the linear space V R^N. Whenever there is no confusion at risk, having …xed the maximal number of jumps M 1; we drop the dependency on M (i.e. we write Vⁿ instead ofV^M;n for all0 n M).

Remark 5 (i) A simple recurrence argument shows that V^M;n V^M;m, for every 0 n m M. Further- more, V^{M;M n} =V^{M0;M0 n}; for all 0 n M M⁰: Moreover, since the dimension of kerB cannot exceed N; V^M;0=V^min(M;Np);0:

(ii) This spaces do not depend on the choice of the controllability horizonT >0:Therefore, if the approximate (null-)controllability is described by these sets, it is independent of the time horizon.

The main result of the paper is the following.

Theorem 6 The switch system (1) is approximately null-controllable (in timeT >0) with 0 as initial mode, if and only if the generated set V⁰₀ reduces tof0g:

The proof is postponed to Section 4. This proof uses the reduction of backward equations with respect to Marked point processes to a system of ordinary di¤erential equations given in [4]. In order to formulate this system (see Proposition 12), we need to explain some concepts and notations in Subsection 4.1. To prove necessity of the condition, one uses convenient feedback controls and the equivalence between invariance and the concept of feedback invariance (see Proposition 13). Su¢ ciency (given by Proposition 14) follows from (time-) invariance of convenient linear subspaces with respect to ordinary di¤erential dynamics.

2.3 Comparison With [13]

We begin with giving a di¤erent (and simpler) proof of (some of) the results in [13]. Besides the general (abstract) characterization of approximate and approximate null-controllability, explicit invariance criteria were given in two speci…c settings.

(i) In the case without multiplicative noiseC = 0; one notes that the subspaces Vⁿ (for0 n < M) do not depend on n:They reduce, in fact, to the largest A ( )-invariant subspace of kerB : Moreover, in this framework,A ( )-invariance andA ( )-invariance coincide and Theorem 6 yields the following.

Criterion 7 ([13, Criterion 4]) The system (1) is approximately null-controllable (with initial mode ₀2E) if and only if the largest subspace of kerB which isA ( ₀)- invariant is reduced to the trivial subspace f0g for all ₀2E:

(ii) Inthe case of Poisson-driven systems with mode-independent coe¢ cientsA andC;one works with the mode-independent operator A :=A R

E(C ( ) +I) Q(d ). The reader familiar with [13, Criterion 3] will note that the necessary and su¢ cient criterion concerns a notion of strict invariance. We get the same condition provided the system has the possibility to stabilize (the maximal number of jumps M N + 1 is allowed to exceed the dimension of the state space). Moreover, without loss of generality, one assumes that E is the support of Q:

Criterion 8 ([13, Criterion 3]) Let us assume that A 2 R^{N N}; B 2 R^{N d} are …xed and C( ) 2 R^{N N}; for all 2 E and that ( )Q( ; d ) is independent of 2 E: Moreover, we assume that M N + 1. Then the associated system is approximately null-controllable if and only if the largest subspace V0 kerB which is (A ; [C ( ) _V0 : 2E])-invariant is reduced tof0g.

Proof. The reader will note that the Vⁿ spaces in (4) no longer depend on 2 E: They are obviously non- decreasing inn(see Remark 5). SincekerB R^N;it follows that, provided thatM N+ 1;one hasV⁰=V¹ (indeed, V⁰ V¹ ::: V^M = kerB and, whenever V^k = V^k+1; for some 0 k M 1, it follows (by de…nition), thatV^j =V^k;for allj k:On the other hand, the inclusions cannot always be strict if M N+ 1 by recalling that the dimension of kerB cannot exceed N). Moreover, this space V⁰ is the largest subspace V₀ kerB which is (A ; [(C ( ) +I) _V0 : 2E])-invariant which is the same as (A ; [C ( ) _V0 : 2E])- invariant. The proof is complete by invoking Theorem 6.

2.4 Approximate or Approximate Null-Controllability

Using Riccati techniques, one proves (see [13, Criterion 3]) that, for Poisson-driven systems with mode-independent coe¢ cients, approximate controllability and approximate null-controllability properties coincide. However, in the case of actual switching systems, the two notions have no reason to and do not coincide. This is illustrated by the following example.

Example 9 We consider the space dimensionN = 4; the control dimensiond= 2; E=f0;1g, a switching rate

= 1 and a transition probabilityQ( ;1 ) = 1;for 2E:Moreover, we consider, for 2 f0;1g;

B = 0 BB

@ 1 0 0 1 0 0 0 0

1 CC

A; A( ) = 0 BB

@

0 0 0 0

0 0 0 0

1 + 0 0 0

0 2 0 0

1 CC

A; C( ;1 ) = 0 BB

@

1 0 0 0

0 1 0 0

0 1 0

0 1 0 1

1 CC A:

The reader is invited to note that kerB = span e³; e⁴ (standard vectors of the basis of R⁴): Thus, simple computations yield V₀¹ span e⁴ ; V₁¹ span e³ : Hence, V₀⁰ =V₁⁰=f0g and the system is approximately null-controllable starting from every initial mode (if M 2). However, if one considers 0 = 0; assumes the mode can jump twice M = 2 and sets := 1T1 T <T2e³ 1T2 Te³; then one easily notes that (Yt; Zt) :=

1T1 t T ;t<T2e³ 1T2 t Te³;(1t T^T1 2 1T1<t T)e³ obey the equation (2). To this purpose, it su¢ ces to note that A ( t)Yt+ (C ( t;1 t) +I)Zt = 0 on [0; T^T2]: For every u 2 U^ad; Itô’s formula (e.g. [15, Chapter II, Section 5, Theorem 5.1]) applied to the inner product D

X^0;u; YE

on[0; T] yields E^0;0hD

X_T^0;u; Ei

= E^{0; 0}hRT

0 hut; B Ytidt i

= 0: In particular, this implies that E^0;0 X_T^{0;u 2} E^0;0 h

j j² i

>0 and, thus, the system (1) is not approximately controllable (towards ).

In fact, the reader may note that the null-controllability property strongly depends on the initial mode (through the computation of V⁰₀ as last step). A su¢ cient criterion (already available in [13]) is that the largest subspace of kerB which is(A ( 0) ; [(C ( 0; ) +I) kerB : 2E; Q( 0; )>0]) invariant should be reduced to f0g:It turns out that asking this condition to hold true for all 02E actually implies approximate controllability. (The proof is postponed to Section 4.)

Condition 10 Let us assume that the largest (A ( ) ; [(C ( ; ) +I) kerB :Q( ; )>0])-invariant subspace of kerB is reduced to f0g, for every 2 E: Then, for every T > 0 and every 0 2 E, the system (1) is approximately controllable in time T >0:

Remark 11 The reader is invited to note that the notion of (A ( ) ; [(C ( ; ) +I) _kerB :Q( ; )>0]) - invariance and that of (A ( ) ; [(C ( ; ) +I) _kerB :Q( ; )>0])-invariance coincide for subspaces ofkerB . Second, according to [13, Criterion 3], the notions of approximate and approximate null-controllability coincide in the context of Poisson-driven systems with mode-independent coe¢ cients. Then, a careful look at [13, Example 4] provides an example of system which is approximately controllable without satisfying the su¢ cient condition given before.

3 Towards Applications

A model. We will explain how the previous method can be applied in the study of stochastic gene networks. To this purpose, we consider the following reaction system describing a repressed bacterium operon model introduced in [19].

D+R^K1DR; D+RN AP ^K2DRN AP; DRN AP !^k3 T rRN AP; T rRN AP !^k4 RBS+D+RN AP RBS!^k5 ?; RBS+Rib^K6RibRBS; RibRBS !^k7 ElRib+RBS; ElRib!^k8 P rotein

P rotein!^k9 F oldedP rotein; P rotein^k!¹⁰?; F oldedP rotein^k!¹¹?:

Partitioning and simplifying. The authors of [5] propose a partition of "species" according to which only ElRib; P roteinandF oldedP roteinare continuous. The averaging procedures in [5, Figure 4] simplify the model to

(5) D

K₃

T rRN AP !^k4 RBS !^k5 ?; RBS !^k7 ElRib+RBS ;

ElRib!^k8 P rotein; P rotein!^k9 F oldedP rotein; P rotein^k!¹⁰?; F oldedP rotein^k!¹¹?:

Due to the conservation law of [D; R; DR; RN AP; DRN AP; T rRN AP] one should have something likeD + T rRN AP '1:

It is known ([5, Page 21]) that "RBS presents infrequent bursts of activity leading to rapid production of ElRib" and "RBS rapidly switches to0by the reactionRBS !?". To take into account these elements and keep the conservation law, we proceed as follows :

(1) asRBS switches to0; D will be reset to1(hence, D +T rRN AP+RBS = 1);

(2) bursts (given by the reaction havingk₇ as speed) will be considered as part of the stochastic updating of the continuous species and will have null-mean (i.e. they will multiply the martingale measure generated by the mode switching mechanism). In our toy-model, asRBS switches to1, stochastic bursts onElRibwill a¤ect (in multiplicative way) the synthesis of P rotein(i.e. the reactionElRib!^k8 P rotein):

A toy mathematical system. The …rst condition leads to a mode spaceE= e¹; e²; e³ consisting of the standard vector basis ofR³;with a jump intensity and a transition measure Q eⁱ; e^j =Q_{i;j 1 i;j 3} given by

(6) ( ) =

*0

@ k₃ k ₃+k₄

k₅ 1 A;

+

>0; for all 2E; Q= 0 B@

0 1 0

k ₃

k ₃+k₄ 0 _k ^k4

3+k₄

1 0 0

1 CA:

We are going to assume that the positive reaction speedsk₇; k8; k9andk11depend on the mode (note thatRBS is part of and intervenes to getElRib)and, maybe, of external one-dimensional control parameters (temperature or catalysts). Since all the reactions concerning continuous components have one reactant, the resulting ODE

will be linear (see [5, Eq. (28)]). A …rst order model for the control will give dxt= [A( t)xt+But]dt;where A is given by (7). Furthermore, in our toy model, let us assume that the external control focuses on regulation of ElRib(i.e. B = 1 0 0 ^t=e¹). We add to that the bursts (see item (2) above) to …nally get a (toy-)model of type (1) for which, for every ; 2E;

(7) B =e¹; A( ) = 0

@ k8( ) 0 0

k8( t) k9( ) 0 0 k9( t) k11( )

1

A; C( ; ) = 0

@ 0 0 0

k₇( ) 0 0

0 0 0

1

A; k₇( ) = 1_e³( ):

Approximate null-controllability. The largest subspace ofkerB which is A e² e² I; kerB - invariant reduces tospan e³ and the largest subspace ofkerB which is A e¹ e¹ I; _span(e³₎ invariant isf0g(recall thatk₈ andk₉are reaction speeds and, thus, are strictly positive and so is ):Due to the structure of the transition measure Q, as soon asM 2, the system is approximately null-controllable starting from e¹: Nevertheless, the spacekerB beingA e³ k₈ e³ k₅ C e³; e¹ +I -invariant, constructions similar to Example 9 show that, providede³ is reachable inM jumps, the system is not approximately controllable.

4 Proof of the Results

4.1 Technical Preliminaries

Before giving the reduction of our backward stochastic equation to a system of ODE, we need to introduce some notations making clear the stochastic structure of several concepts : …nal data, predictable and càdlàg adapted processes and compensator of the initial random measure. The notations in this subsection follow the ordinary di¤erential approach from [4]. Since we are only interested in what happens on[0; T];we introduce a cemetery state (1; ) which will incorporate all the information after T ^TM: It is clear that the conditional law of T_n+1 given (T_n; _Tn) is now composed by an exponential part on[T_n^T; T] and an atom at1: Similarly, the conditional law of _Tn+1 given (T_n+1; T_n; _Tn) is the Dirac mass at if T_n+1 = 1 and given byQ otherwise.

Finally, under the assumptionP^{0; 0}(T_M+1=1) = 1, afterT_M;the marked point process is concentrated at the cemetery state.

We setET : = ([0; T] E)[ f(1; )g. For everyn 1; we let ET ;n ET

n+1 be the set of all marks of typee= ((t0; 0); :::;(tn; n));where

(8) t0= 0; (ti)_{0 i n} are non-decreasing; ti< ti+1; ifti T; (ti; i) = (1; ), ifti > T; 80 i n 1;

and endow it with the family of all Borel sets Bn. For these sequences, the maximal time is denoted byjej:=t_n. Moreover, by abuse of notation, we set _jej:= _n:WheneverT t >jej;we set

(9) e (t; ) := ((t0; 0); :::;(tn; n);(t; ))2ET ;n+1: By de…ning

(10) e_n:= ((0; ₀);(T₁; _T1); :::;(T_n; _Tn)); we get anET ;n valued random variable, corresponding to our mode trajectories.

The …nal data is F[0;T] measurable and, thus, for every n 0; there exists aBⁿ=B R^N measurable functionET ;n3e7! ⁿ(e)2R^N such that:

(11) If jej=1; then ⁿ(e) = 0: Otherwise, onT_n(!) T < T_n+1(!); (!) = ⁿ(e_n(!)):

A càdlàg processY continuous except, maybe, at switching times T_n is given by the existence of a family of Bⁿ B([0; T])=B R^N -measurable functions yⁿ such that, for alle2ET ;n; yⁿ(e; )is continuous on [0; T]and constant on[0; T ^ jej]and

(12) If jej=1; then yⁿ(e; ) = 0:Otherwise, onTn(!) t < Tn+1(!); Yt(!) =yⁿ(en(!); t); t T.

Similar, an R^N valued F-predictable process Z de…ned on [0; T] E is given by the existence of a family ofBn B([0; T]) B(E)=B R^N measurable functionszⁿ satisfying

(13) If jej=1; thenzⁿ(e; ; ) = 0:OnTn(!)< t Tn+1(!); Zt(!; ) =zⁿ(en(!); t; ); fort T, 2E.

To deduce the form ofthe compensator, one simply writesqb(!; dt; d ) :=X

n 0

b qⁿ_e

n(!)(dt; d ) 1_{Tn(!)<t Tn+1(!)^T} such that

(14)

( Ifn M; thenqbⁿ_e(dt; d ) = (d ) ₁(dt). Ifn M 1;

b

q_eⁿ(dt; d ) := ( _jej)Q( _jej; d )1_{jej<1;t2[jej;T]}Leb(dt) + (d ) ₁(dt) 1_{(jej<1;t>T)[jej=1}; :

Let us now concentrate on the speci…c form of the jump contribution Z (to the BSDE (2)). We consider a càdlàg process Y continuous except, maybe, at switching times Tn. Then, as explained before, this can be identi…ed with a family (yⁿ):We construct, for everyn 0;

(15) byⁿ⁺¹(e; t; ) :=yⁿ⁺¹(e (t; ); t) 1_jej<t

and YTn+1 can be obtained by simple integration of the previous quantity with respect to the conditional law of Tn+1; Tn+1 knowingF^Tn:Then,Z is given byzⁿ(e; t; ) :=ybⁿ⁺¹(e; t; ) yⁿ(e; t):

Thecoe¢ cient function A( t)is adapted and can be seen as follows: if jej =1; then A= 0; otherwise, one works with A _jej : Similar constructions hold true for C: In fact, the results of the present paper can be generalized to more general path-dependence of the coe¢ cients.

4.2 Reduction to a System of Linear ODEs

We consider the family of (ordinary) di¤erential equations

(16) 8>

>>

><

>>

>>

:

y^M(e_M(!); ) = ^M(e_M(!)). Forn M 1; yⁿ(e_n(!); T) = ⁿ(e_n(!)); dyⁿ(en(!); t) = AR _jen(!)j yⁿ(en(!); t)dt

E C _jen(!)j; +I byⁿ⁺¹(en(!); t; ) yⁿ(en(!); t) qbⁿ_e

n(!)(dt; d ) (=PA jen(!)j yⁿ(e_n(!); t)dt

2E ( _jen(!)j)Q( _jen(!)j; ) C _jen(!)j; +I yⁿ⁺¹(e_n(!) (t; ); t)dt);

where we have used the notation (3). The following result adapts [4, Lemma 7] to our case.

Proposition 12 A càdlàg adapted processY given by a family of functions(yⁿ) as in (12) is solution to (2) if and only if, for P-almost all ! and all0 n M; it satis…es the system (16).

The proof is quasi-identical to the one of [4, Lemma 7]. The only di¤erence in our case is the presence of the term A _jen(!)j yⁿ(e_n(!); t)dt which is, of course, classical. The results of [4, Lemma 7] apply directly if one assumes that ( ) >0 for all 2E (that is if there exists no absorbing state). Otherwise, we actually get an ODE of typedyⁿ(e_n(!); t) = A _jen(!)j yⁿ(e_n(!); t)dt:

4.3 An Iterative Invariance-Based Criterion (Proof of Theorem 6)

As already hinted in [13], the (approximate) controllability properties can be expressed with respect to invariance conditions. The equivalence between the dual (backward) stochastic equation (2) and the (backward) ordinary di¤erential system (16) yields the following approximate controllability criterion.

Proposition 13 If the system (1) is approximately null-controllable with 0 as initial mode, then the generated set V⁰₀ reduces tof0g:

Proof. Using classical results on the di¤erent notions of invariance (e.g. [22, Theorem 3.2], see also [6, Lemma 4.6]), invariance is equivalent to feedback invariance. Thus, one gets the existence of a family of operators Fⁿ_; 2 L Vⁿ;Vⁿ⁺¹ such that A ( ) +P

2E; Q( ; )>0(C ( ; ) +I)Fⁿ_; Vⁿ Vⁿ; for alln 0: We begin

with picking (an arbitrary) v₀ 2 V⁰₀ and de…ne ⁰(t₀; ₀) = v₀: We proceed by setting, for every n 1 and en 2ET ;n, n;en to be the unique solution of the ordinary di¤erential system

8>

>>

>>

<

>>

>>

>:

d _n;en(t) = A jenj +P

2E,Q( n ; )>0 C ( _jenj; ) +I Fⁿ

jenj; n;en(t)dt; je_nj t T

n;en(jenj) =

n(en); if jenj<1, 0; otherwise. and

n+1(e_n (t; )) =

1

( _jenj)Q( _jenj;)Fⁿ

jenj; n(e_n; t) 1_{T t>jenj}; if ( _jenj)Q( _jenj; )>0;

0; otherwise

One also sets n;en(t) = n;en(jenj _t) to extend the solution for t 2 [0; T]. Then, one easily notes that

n;en(t) 2 kerB ; for all 1 n M, all en 2 ET ;n and all t 2 [0; T]: Moreover, a simple glance at the construction shows that by setting yⁿ(en; t) := n;en(t), for1 n M, allen 2ET ;n and all t2 [0; T]; one gets the solution of (16) with the particular choice of the …nal data such that ⁿ(en) = n;en(T): Since we have assumed the system (1) to be approximately null-controllable, Theorem 2 and Proposition 12 yieldv₀= 0:

The proof is complete by recalling thatv₀2V⁰₀ is arbitrary.

At this point, the reader may want to note that these considerations involve one equation at the time. The invariant space obtained is then employed for the next equation and gives a coherent character to the system.

The basic idea is to provide some kind of local in time invariance of the sets concerned. In [13], this is done using Riccati techniques. But, except for special cases, the solvability of these stochastic schemes is far from obvious.

Due to the ordinary di¤erential structure of the equivalent system (16), we are able to elude these techniques and work directly on the deterministic systems.

Proposition 14 Conversely, if the generated set V⁰₀ reduces tof0g; then the system (1) is approximately null- controllable with ₀ as initial mode.

Proof. We begin with a solution of (2) for whichY belongs to kerB :We prove by descending recurrence that yⁿ(e; t) 2 Vⁿ

jej; for all t 2 [0; T] and all e 2 E_{T ;n} (starting from ₀), where we use the structure (12): The assertion is obvious for n=M since, by notation, V^M = kerB :We assume it to hold true forn+ 1 M and prove it for n 0:By equation (16), one has

dyⁿ(e; t) = A jej yⁿ(e; t) X

2E

( _jej)Q( _jej; ) C _jej; +I yⁿ⁺¹(e (t; ); t)

! dt:

We have assumed that yⁿ(e; t)2kerB and, thus,[I kerB ]yⁿ(e; t) = 0:We infer that A jej yⁿ(e; t) +X

2E

( _jej)Q( _jej; ) C _jej; +I yⁿ⁺¹(e (t; ); t)2kerB :

Hence, using the recurrence assumption,yⁿ(e; t)is (for almost allt2[0; T]);an element of the linear space

W⁰:=

8<

:

v2kerB :9w 2Vⁿ⁺¹; for all 2E s.t. Q _jej; >0 satisfying A jej v+ P

2E; Q( jej; )^>0

C _jej; +I w 2kerB

9=

;:

By repeating our argument, we prove thatyⁿ(e; t)is (for almost allt2[0; T]);an element of the linear space

W^m+1:=

8<

:

v2W^m:9w 2Vⁿ⁺¹; for all 2Es.t. Q _jej; >0 satisfying A jej v+ P

2E; Q( jej; )^>0

C _jej; +I w 2W^m

9=

;;

for everym 0:Then,W := \

0 m NW^mis an A jej ;h

C _jej; +I _Vⁿ⁺¹ :Q _jej; >0i

invariant subspace of the (at mostN-dimensional) spacekerB :Therefore, we have proven thatyⁿ(e; t)2Vⁿ

jej:To complete our argument, one only needs to recall that, by assumption, V⁰₀ =f0g and use Theorem 2 and Proposition 12.

4.4 Proof of Su¢ ciency Condition 10 for Approximate Controllability

Proof of Condition 10. In light of the Theorem [13, Theorem 1] and Proposition 12, one only needs to show that the only solution of (16) remaining inkerB is constant0:One proceeds as in the Proof of Proposition 14 starting with a solution of (2) for which Y belongs tokerB and showing that yⁿ(e; t)2Vⁿ

jej kerB ;for allt2[0; T] (recall that yⁿ is continuous). One recalls thatVⁿ is A ( ) ;

h

(C ( ; ) +I) _Vⁿ⁺¹ :Q( ; )>0 i

-invariant, for every 2E. Hence, a fortiori, Vⁿ

jej is A jej ; C ( _jej; ) +I kerB :Q _jej; >0 -invariant. Our assumption implies that Vⁿ

jej =f0gand approximate controllability follows.

References

[1] V. Barbu, A. R¼a¸scanu, and G. Tessitore. Carleman estimates and controllability of linear stochastic heat equations. Appl. Math. Optim., 47(2):97–120, 2003.

[2] Pierre Brémaud. Point processes and queues : martingale dynamics. Springer series in statistics. Springer- Verlag, New York, 1981.

[3] R. Buckdahn, M. Quincampoix, and G. Tessitore. A characterization of approximately controllable linear stochastic di¤erential equations. In Stochastic partial di¤ erential equations and applications— VII, volume 245 of Lect. Notes Pure Appl. Math., pages 53–60. Chapman & Hall/CRC, Boca Raton, FL, 2006.

[4] F. Confortola, M. Fuhrman, and J. Jacod. Backward stochastic di¤erential equations driven by a marked point process: an elementary approach, with an application to optimal control.Annals of Applied Probability, to appear, 2015. arXiv:1407.0876.

[5] A. Crudu, A. Debussche, and O. Radulescu. Hybrid stochastic simpli…cations for multiscale gene networks.

BMC Systems Biology, page 3:89, 2009.

[6] R. F. Curtain. Invariance concepts in in…nite dimensions. SIAM J. Control and Optim., 24(5):1009–1030, SEP 1986.

[7] M. H. A. Davis. Markov models and optimization, volume 49 of Monographs on Statistics and Applied Probability. Chapman & Hall, London, 1993.

[8] E. Fernández-Cara, M. J. Garrido-Atienza, and J. Real. On the approximate controllability of a stochastic parabolic equation with a multiplicative noise. C. R. Acad. Sci. Paris Sér. I Math., 328(8):675–680, 1999.

[9] D. Goreac. A Kalman-type condition for stochastic approximate controllability. Comptes Rendus Mathema- tique, 346(3–4):183 –188, 2008.

[10] D. Goreac. Approximate controllability for linear stochastic di¤erential equations in in…nite dimensions.

Applied Mathematics and Optimization, 60(1):105–132, 2009.

[11] D. Goreac. A note on the controllability of jump di¤usions with linear coe¢ cients. IMA Journal of Mathe- matical Control and Information, 29(3):427–435, 2012.

[12] D. Goreac. Controllability properties of linear mean-…eld stochastic systems. Stochastic Analysis and Ap- plications, 32(02):280–297, 2014.

[13] D. Goreac and M. Martinez. Algebraic invariance conditions in the study of approximate (null-) controlla- bility of Markov switch processes. Math. Control Signals Systems, 27(4):551–578, 2015.

[14] M. L. J. Hautus. Controllability and observability conditions of linear autonomous systems. Nederl. Akad.

Wetensch. Proc. Ser. A 72 Indag. Math., 31:443–448, 1969.

[15] N. Ikeda and S. Watanabe. Stochastic Di¤ erential Equations and Di¤ usion Processes, volume 24 of North- Holland Mathematical Library. North-Holland Publishing Co., Amsterdam–New York; Kodansha, Ltd., Tokyo, 1981.

[16] B. Jacob and J. R. Partington. On controllability of diagonal systems with one-dimensional input space.

Systems & Control Letters, 55(4):321 –328, 2006.

[17] B. Jacob and H. Zwart. Exact observability of diagonal systems with a …nite-dimensional output operator.

Systems & Control Letters, 43(2):101 –109, 2001.

[18] Martin Jacobsen. Point Process Theory And Applications. Marked Point and Piecewise Deterministic Processes. Birkhäuser Verlag GmbH, 2006.

[19] Sandeep Krishna, Bidisha Banerjee, T. V. Ramakrishnan, and G. V. Shivashankar. Stochastic simulations of the origins and implications of long-tailed distributions in gene expression. Proceedings of the National Academy of Sciences of the United States of America, 102(13):4771–4776, 2005.

[20] S. Peng. Backward stochastic di¤erential equation and exact controllability of stochastic control systems.

Progr. Natur. Sci. (English Ed.), 4:274–284, 1994.

[21] D. Russell and G. Weiss. A general necessary condition for exact observability. SIAM Journal on Control and Optimization, 32(1):1–23, 1994.

[22] E. J. P. G. Schmidt and R. J. Stern. Invariance theory for in…nite dimensional linear control systems.Applied Mathematics and Optimization, 6(2):113–122, 1980.

[23] M. Sirbu and G. Tessitore. Null controllability of an in…nite dimensional SDE with state- and control- dependent noise. Systems & Control Letters, 44(5):385–394, DEC 14 2001.