On the optimal control of linear complementarity systems

Transcription

1 On the otimal control of linear comlementarity systems Alexanre Vieira, Bernar Brogliato, Christohe Prieur To cite this version: Alexanre Vieira, Bernar Brogliato, Christohe Prieur. On the otimal control of linear comlementarity systems. IFAC Worl Congress, Jul 27, Toulouse, France. HAL I: hal htts://hal.inria.fr/hal-49388v2 Submitte on 24 Mar 27 HAL is a multi-iscilinary oen access archive for the eosit an issemination of scientific research ocuments, whether they are ublishe or not. The ocuments may come from teaching an research institutions in France or abroa, or from ublic or rivate research centers. L archive ouverte luriiscilinaire HAL, est estinée au éôt et à la iffusion e ocuments scientifiques e niveau recherche, ubliés ou non, émanant es établissements enseignement et e recherche français ou étrangers, es laboratoires ublics ou rivés.

2 On the otimal control of linear comlementarity systems Alexanre Vieira Bernar Brogliato Christohe Prieur INRIA Grenoble Rhône-Ales, Univ. Grenoble-Ales, 655 avenue e l Euroe, Saint-Ismier, France, alexanre.vieira@inria.fr - bernar.brogliato@inria.fr) Univ. Grenoble Ales, CNRS, GIPSA-lab, F-38 Grenoble, France christohe.rieur@gisa-lab.fr) Abstract: In this aer, we give some results concerning the quaratic otimal control of linear comlementarity systems. We erive first orer conitions for this system, that we manage to exress as a Mixe Linear Comlementarity System. We then use this result to buil numerical schemes, which are exresse as Mathematical Problems with Equilibrium Constraints. Keywors: Linear otimal control, Comlementarity roblems.. INTRODUCTION We are intereste in ynamical control systems having the form: { ẋt) Axt) + Bλ x t) + F ut), yt) Cxt) + λ x t) + eut), x) x, xt ) free a.e. on [, T ] ) where A R n n, x ), B, F R n, C R n,, e R, >, T >, an u, λ x : [, T ] R are measurable. In orer to avoi trivial cases, we assume that C, e), ). Furthermore, the trajectory of this system has to comly with the following comlementarity conitions: λ x t) yt) 2) meaning that λ x t),yt) an yt)λ x t) for almost all t [, T ]. The whole system )-2) is calle a controlle linear comlementarity system CLCS). This kin of systems, esite their simle look, gives rise to several challenging questions, mainly because conitions 2) introuce non-ifferentiability at switching oints an non-convexity of the set of constraints. It rovies a moeling araigm for many roblems, as Nash equilibrium games, hybri engineering systems Brogliato 23)), contact mechanics or electrical circuits Acary et al. 2)). Several roblems have alreay been tackle, let us mention observer-base control Çamlibel et al. 26), Heemels et al. 2)) an Zeno behavior Çamlibel an Schumacher 2), Pang an Shen 27), Shen 24)). We efine now the otimal control roblem of fining the trajectories of )-2) minimizing the functionnal: Cu) T f xt), ut))t T xt) xt) + ut) 2 )t. 3) The otimal control of this system is a really challenging question. For instance, the existence of an otimal control still is an oen fiel of research. A well-known theorem, ue to Filliov see Cesari, 22, Theorem 9.2i an onwars), requires the convexity of the set Ux) {u, v) : v Cx + Dv + Eu } which, in this framework, is actually not convex. Another ifficulty comes from the fact that the constraints involve both the control an the state. These mixe constraints make the analysis even more challenging. For instance, eriving a maximum rincile with wie alicability involves the use of non-smooth analysis, even in the case of smooth an/or convex constraints see e.g. Clarke an De Pinho 2)). Secial cases arise when yt) Cxt) + λ x t) meaning e in )). This system can then be seen as an autonomous switching system, where the switching moes are activate when the state reaches some threshol efine by the comlementarity conitions 2) see Georgescu et al. 22)). The otimal control of such systems has been alreay stuie see Passenberg et al. 23) an references therein). Since the control u is also involve in the constraints 2), all these results o not aly. However, in our moel, we can get ri of the constraints an comute exlicitly λ x as a function of u an x in a rather simle form. Nonetheless, this exression being non ifferentiable, we nee owerful tools from non-smooth analysis in orer to erive a Pontryaginlike maximum rincile. The contribution of this aer is that we manage to exress these first-orer conitions as a Mathematical Problem with Equilibrium Constraints MPEC) an buil on it a numerical scheme. Since the control u was not involve in the constraints in the aforementione litterature, this aroach is new. This aer is organize as follows: the first section is evote to the erivation of results leaing to a maximum rincile for this system. We then recall some results on comlete controllability systems, an erive numerical schemes to obtain a numerical solution. We then analyze a simle one-imensional examle, which serves as a benchmark for the numerical schemes. Conclusion ens the aer in Section PRELIMINARY RESULTS Moel )-2) is rewritten as follows:

3 { ẋt) Axt) + Bλ x t) + F ut), λ x t) Cxt) + λ x t) + eut), a.e. on [, T ] x) x, xt ) free 4) In this case, basic convex analysis roves that λ x t) Π R + Cx eu), where Π Kx) is the rojection of x on the set K. Therefore, 4) becomes where the argument t is omitte for simlification): ẋ f x x, u), Ax + F u + B Π R + Cx eu), Ax + F u if Cx + eu, A BC ) x + F e ) B u otherwise, 4 ) with obvious efinition for f x. In 3), the function uner the integral sign is smooth, so its subifferential only contains the classical erivative: x f x, u) {2x }. The vector fiel of the ynamical control system is non smooth, but we can comute its subifferential as: { {A} if Cx + eu >, x f x A BC } if Cx + eu <, x, u) [ A BC ], A if Cx + eu. using the notation: [M, M 2 ] conv{m, M 2 } for any air M, M 2 ) of accoring imensions matrices an conv stans for the convex hull of M an M 2. As state by Clarke 976), if the control, an therefore the trajectory, is otimal, then there exist an absolutely continuous function : [, T ] R n an a scalar such that, ) an satisfying a Pontryagin-like ifferential inclusion: ṗ t) f x,, u) t) x f x xt), ut)) + x f xt), ut)). 5) In our roblem, 5) becomes: {2 { x t) + t)a} if Cx + eu >, 2 x t) + t) A BC )} ṗ t) [ if Cx + eu <, {2 x t)} + t) A BC ], A if Cx + eu. 6) Furthermore, the maximum conition on the Hamiltonian hols: t), f x xt), ut)) + f xt), ut)) max v R { t), f x xt), v) + f xt), v)}. 7) Since xt ) is free, there is a terminal conition on : T ). 3. COMPLETE) CONTROLLABILITY CONDITIONS In orer to comute an otimal control, the system shoul obviously be controllable between the inital an final oints. We will only focus on comletely controllable systems, relying on Çamlıbel s theorem stating the comlete controllability of some LCS. We recall here his result: Theorem. Çamlibel 27)). Assume 4) satisfies the following conitions with D an E e): ) The matrix D is a P -matrix ; i.e., all its rincial minors are ositive. 2) The transfer matrix E + CsI A) F is invertible as a rational matrix. Then, 4) is comletely controllable if, an only if, the following two conitions hol: ) The air A, [F B]) is controllable. 2) The system of inequalities η, 8a) ) ζ η A λi F ), 8b) C E ζ η B ), 8c) D) amits no solution λ R an ζ, η) R n+m. This will be use in Section 6 on a articular case. 4. 4) AND 6) AS A MIXED LCS MLCS) We enote z x ). Let us rewrite 4) an 6) in the comact form: ) f ż x z, u) f 9) z, u) where the right-han sie is a set-value function efine by: if Cx + eu > : ) f x z, u) f z, u) { )} Ax + F u 2 x A, if Cx + eu < : ) f x z, u) A CB ) x + F e ) f B u z, u) C 2 B ) x + A, if Cx + eu : ) f x z, u) A F C ) x f e [ z, u) C 2 B ] x + A, A. We can recast the ifferential inclusion 9) in the framework of comlementarity systems with linear ynamics: ) ) g ż x z, u) A g z, u) 2 I A z ) B λ ) ) + C B x F λ + u, )

4 λ x Cx + λ x + eu, a) λ j Cx + eu + Cx + eu, b) λ j 2 Cx + eu Cx + eu) j...n, c) λ j + λj 2 j j...n, ) λ + λ 2, e) where the subscrit j enotes the j-th comonent of a vector, an λ i λ i,..., λ n n ). Proosition 2. The right-han sie of 9) is the same as the right-han sie of ) efine with the comlementarity conitions in ). Proof. The first line a) gives obviously the same righthan sie as in 4), which is f x z, ). Then, f x z, u) g x z, u). Therefore, we have to check that the other lines in ) are the same as in 6). To o so, we have to istinguish 3 cases: if Cx + eu >, then from b), we euce that λ. It follows that: g z, u) 2 x A f z, u). if Cx+eu <, then from c), λ 2, an therefore from e), λ. We then have the following equality: C g z, u) 2 B ) + A f z, u). if Cx + eu, then from ), we have that λ j + λj 2 j, so that λ j i [ j, j ], i, 2, j,...n. But, in orer to comly with the last equality e), we must have λ j i [, ] j, i, 2, j,..., n, an hence, λ i [, ], i, 2. This gives us g z, u) { 2 x A } + [, ] C B [ C { 2 B ] x} + A, A f z, u). We can still have an even more interesting form of ) by noticing that the function is iecewise linear an so, amits a reresentation in the form of an LCP. This is the toic of the next lemma: Lemma 3. Define n,..., ) R n. The multiliers λ x an λ given by system ) are equally efine by the following system: λ x Cx + λ x + eu µ x,u µ x,u 2Cx + eu) µ µ 2 µ x,u n 2 µ x,u 2Cx + eu)) n µ µ 2λ 2) µ 2 µ 2 2λ µ λ + λ 2 µ λ λabs µ 2 λ 2 λabs 2 where 2) is a mixe LCP MLCP). Proof. First, we nee to establish the following simle result: for any scalar x, x µ x where µ is given by: µ µ 2x. Inee, if x, then we must take µ, so that µ x x x. If x >, then we must take µ 2x, an µ x x x. Let us use ) to rewrite equivalently the absolute values as: µ x,u µ x,u 2Cx + eu), µ µ 2λ, µ 2 µ 2 2λ 2, µ µ 2, x,u µ x,u Cx + eu) Cx + eu, µ λ λ, 2 µ 2 λ 2 λ 2, µ, where notation x on a vector x is here unerstoo comonentwise, e.g. x x,..., x n ). Therefore, ) becomes: λ x Cx + λ x + eu µ x,u µ x,u 2Cx + eu) µ µ 2λ µ 2 µ 2 2λ 2 µ µ 2 Cx + eu + x,u) n 2 x,u Cx + eu)) n + 2 λ + λ 2 x,u µ x,u Cx + eu) µ λ 2 µ 2 λ 2 µ Noticing that we can use the two equalities on x,u an an insert them above, we have roven that λ x an are equally efine by ) or 2). λ Therefore, we infer that the right-han sie of the ifferential inclusion in 9) is equal to the right-han sie of system ): ż Ãz + BΛ + F u, 3) where Ã, B, F an Λ are easily ientifiable from ), an subject to the MLCP 2). 5. NUMERICAL OPTIMAL SOLUTION 5. Direct metho A first way to solve the otimal roblem is to iscretize irectly the ynamics 4) an the cost 3) in orer to obtain a constraine otimization roblem. In fact, a simle iscretization that we can use here is the following one: min u R N N f x i, u i ), i

5 { λ x k Cx k+ + λ x k + eu k s.t. x k+ x k Ax k+ + Bλ x k...n, k + F u k h where the subscrit k in z k enotes the k-th ste in the iscretization of the variable zt) at time t k, an h enotes the here uniform) time-ste. Moreover, we can choose to integrate the ynamics with an imlicit as resente here) or an exlicit metho. This is a Mathematical Program MP) constraine by a Mixe Linear Comlementarity Problem, written as MLCPG, u, ), H, u, )) with G : R nn+) R N+ R N + R nn+) an H : R nn+) R N+ R N + R N, where G k x, u, λ) x k+ x k Ax k+ Bλ x k F u k, h H k x, u, λ) Cx k+ + λ x k + eu k, k,..., N. We notice that we can isolate x k+ in the ynamics, an reintrouce it in the comlementarity conitions: λ x k + hci ha) B)λ x k+ e + hci ha) F )u k + CI ha) x k As we can see here, this metho is rather simle to imlement, an can integrate further constraints. However, regaring imensions, we see that a small iscretization ste h > will increase ramatically the size of the system to be solve. Nonetheless, the solution of this roblem can be the initial guess of the inirect metho that we will resent in the next section. 5.2 Inirect metho A secon way to comute an aroximate solution of the otimal control roblem is to iscretize 7) with 3)-2). This metho is known as the inirect metho since it is using an a riori stuy of the system, an the result obtaine with the Pontryagin equations). Obviously, the choice of the iscretization will eventually have an imact on the accuracy an the stability of the numerical solution. So as to have a first iea of the extent of these issues, these equations are iscretize with an Euler scheme which will use imlicit or exlicit terms in its formulation, an we will investigate how these choices affect the solution. We resent two formulations, that we name of exlicit an imlicit tye. As we will see, these two formulations lea to ifferent tyes of otimization roblems. Exlicit tye This first formulation leas to a roblem where we can ientify two almost) ineenant roblems at each ste. Let us assume we alreay know variables values of x an at time t k, e.g. z k, an we want to comute the solution at time t k+. We first solve the following iscretization of 7) with 2): max v R { k, f x x k, v) + f x k, v)}, λ x k Cx k + λ x k + ev µ x,u µ x,u 2Cx k + ev) µ µ 2 k µ x,u n 2 µ x,u 2Cx k + ev)) n s.t. µ µ 2λ k µ 2 µ 2 2λ 2k + 2 µ k λ k + λ 2k k µ λ k λabs µ 2 λ 2k λabs 2 Solving this roblem will give us u k an the associate Λ k, which is unique for a given u k as we seen from the erivation of system 2), excet in the case where Cx k + eu k. We can rewrite the comlementarity conitions of this MPEC in the following comact form : Ω Ω + Ψ where Ω k λ x k, µ x,u, µ,, 2, µ, µ 2 ), n n n n n 4) an Ψ is easily ientifiable. Finally, we just nee to integrate the ynamics 3), which is integrate with the following iscretization: z k+ z k h Ãz k+ + BΛ k + F u k. Here again, we can use an imlicit integration, or an exlicit one by introucing z k instea of z k+ in the righthan sie. Imlicit tye This secon formulation is exresse in the form of a single Mathematical Program with Equilibrium Constraints MPEC) solve at each timeste. Here, every variable will be use imlicitly, as the ynamics is introuce insie the constraints of the MP. Namely, we nee to solve at each ste the following MPEC: max { k+, fx k+, v) + f x k+, v)} v R λ x k+ Cx k+ + λ x k+ + ev µ x,u µ x,u 2Cx k+ + ev) µ µ 2 k+ µ x,u n 2 µ x,u 2Cx k+ + ev)) n µ µ 2λ k+ s.t. µ 2 µ 2 2λ 2k+ + 2 µ k+ λ k+ + λ 2k+ k+ µ λ k+ λabs z k+ z k h µ 2 λ 2k+ λabs 2 Ãz k+ + BΛ k+ + F v Avantages an flaws of the inirect methos The main avantage of this kin of methos is that they rouce usually highly accurate solutions Rao, 29). Also, in the framework eveloe here, a really small timeste will only increase the number of maximisation roblems solve, but

6 not the imension of each of these, which is clearly an avantage over the irect metho. However, the otimal control is comute as an oen loo, an we have only necessary conitions but not sufficient ones, such that we still have to check that the solution that is foun, is really otimal. On to of that, even though the systems are quite small, a maximisation roblem will be solve at each ste, which is a comutationationally har roblem. 6. D EXAMPLE In orer to illustrate the revious results, we now focus on a D examle, written in the following generic form: T minimize xt) 2 + ut) 2) t, ẋt) axt) + bλt) + fut), such that: λt) λt) + eut), a.e. on [, T ] x) x, xt ) free, 5) where all variables are scalars, >, b, e. 6. Comlete controllability conitions for this D examle In orer to analyze this roblem, we will first secify the necessary an sufficient comlete controllability conitions in the D case. Alying theorem, we must check that the following system: has no solution λ R an ζ, η) a λ)ζ + cη, 6) fζ + eη, 7) η, 8) bζ + η, 9) If e > : we euce through 7): η fζ e. ) If f, then η. In 6), we can take λ a. However, with 9), we have that ζb. Let us take ζ signb). Then we foun a solution with ζ : the system is not comletely controllable. 2) If f <, then with 8), we have that ζ. Through 6), we take λ a + cf e. If b, then 9) is a sum of ositive terms which must be nonositive, so η an ζ : the system is comletely controllable. If b <, then 9) becomes ζb f e ) with ζ. If b f e then we can take any ζ : the system is not comletely controllable. Otherwise, only ζ suits, so η, an then the system is comletely controllable. 3) If f >, then in 6), we take λ a + cf e. Through 8), we have that ζ. If b, then 9) is a ositive terms sum which must be nonositive, so η an ζ : the system is comletely controllable. If b >, then 9) becomes ζb f e ) with ζ. If b f e then then we can take any ζ : the system is not comletely controllable. Otherwise, only ζ suits, so η, an then the system is comletely controllable. If e < : we have the same cases as with e > by inverting the sign of f. 6.2 Search for the exlicit otimal solution The ynamic system in 5) can be rewritten as: ẋ ax + fu + b Π R + eu). 2) Therefore, the Hamiltonian function is written as: Hx,, u) ax + fu + b ) Π R + eu) x 2 + u 2). 2 2) Ajoint equation We notice that this equation is smooth in x. Therefore, using 6), the ajoint equation is smooth, an is written as: ṗt) at) + xt). We can even ifferentiate twice, an obtain the following secon-orer ifferential equation: aṗ + ẋ a 2 + fu + b Π R + eu) a 2 + fu if eu a 2 + f be ) u if eu. Maximization of the Hamiltonian function We now search for an exression of the otimal control u, function of x an, maximizing the Hamiltonian function Hx,, u ). To that aim, we use the subifferential of H with resect to u, written u Hx,, u), an the fact that if u maximizes H, then u Hx,, u ). In our roblem, the subifferential is written as {f { u} if eu >, f eb ) } u if eu <, u Hx,, u) [ f eb ], f if eu. We now only focus on the comlete controllable cases in orer to fin a control u maximizing this function: If e > : In that case, sgneu) sgnu). ) We consier first f <. If b > ), then if, then f, f eb, an if, then f, ) f eb. We also notice that [ ] f, f eb. So we have: u f if, if.

7 If b <, then we must make sure that f < eb. We notice that in this case, [ ] f, f eb. We are then in the exact same case as the revious one, an therefore, the control is exresse the same way: u f if, if. 2) We consier now f >. If b < ), then if, then f, f eb, an if, then f, ) f eb. We also notice that [ ] f, f eb. So we have: u f if, if. If b >, then we must make sure that f > eb. We notice that in this case, [ ] f, f eb. We are then in the exact same case as the revious one, an therefore, the control is exresse the same way: u f if, if. If e < : we have the same cases as with e > by inverting the sign of f. Therefore, we can summarize this result as follows: f u if ef, if ef. 22) Final ajoint equation Finally, we use the otimal control foun in 22) in the equation foun on. Surrisingly, we en u with a rather simle equation: a 2 + f 2) if ef, a 2 + f be ) ) 2 if ef, which we rewrite in the more simle form: γ) 23) with γ) > an iecewise constant. Initial conitions We nee now to fin ) such that T ) since xt ) is free, accoring to the maximum rincile). On to of that, we know that the initial value for the erivative ṗ is given by: ṗ) x) a) The hase ortrait is eicte in Figure. It is clear that, in orer to have T ), the sign of ) is etermine by the sign of the constants in the moel: If a >, x) >, then T ) ) <, If a >, x) <, then T ) ) >, If a <, x) >, then T ) ) <, If a <, x) <, then T ) ) >. We can summarize this by: sgn)) sgnx)) Fig.. Phase ortrait of 23) - a, b.5,, e 2, f 3, x), On to of that, will always have the same sign on [, T ], so the otimal control u given in equation 22) always has the same sign on [, T ], an is smooth since is smooth). Furthermore, γ will be constant on [, T ]. Consequently, we know exlicitly the solution t) on [, T ], namely: t) 2 γ [ γ a)e γt + ) γ + a)e γt ) + e ) ] γt e γt x). In orer to have T ), we must take: x) e 2 γt ) ) γ a)e 2 γt + γ + a. From that, it is easy to have the exression of the otimal trajectory x, using the fact that xt) ṗt) + at). 6.3 Numerical results Direct metho The irect metho gives rather goo results, whatever the arameters in 5) use. We use the solver GAMS available at htt:// which inclues a owerful MPEC solver. The only trouble notice was that some fluctuations aroun the analytical solution were foun. Nonetheless, these fluctuations are still amissible in all the calculations we mae. Some results are shown in Figure 2. Inirect metho The inirect metho gives more isaointing results. As shown in Figure 3, even with the goo initial value ), this aroach fails to give a goo solution, close to the analytical one. Hoe for a better solution with this metho seems small since in general alication, ) shoul be foun numerically. Here again, GAMS was use to solve the MPEC at each ste. The reason why this is not working still is unknown to us: changing the arameters oes not seem to enhance the recision of the numerical solution, nor the reuction of the timeste h. At each time ste, the resolution of the MPEC seems to fail, the constraints being often largely violate. A first way to exlain this may be foun in matrix introuce in 4) : even in this scalar examle, it is of

8 State : Analytical solution Numerical solution Exlicit) Numerical solution Imlicit) State : Analytical solution.8 Numerical solution Exlicit) Numerical solution Imlicit) Control : Analytical solution Numerical solution Exlicit) Numerical solution Imlicit) Control : Analytical solution Numerical solution Exlicit) Numerical solution Imlicit) Fig. 2. Numerical solution: irect aroach - a, b.5,, e 2, f 3, x), N 6 rank 5, when is 7 7. Further investigations are neee an are ongoing. 7. CONCLUSION The results we foun for the otimal control of linear comlementarity systems are romissing an we were able to formulate the first-orer conition in a convenient framework using comlementarity conitions. Even though we achieve getting these conitions uner a suitable form, the numerical schemes erive from the inirect metho are not satisfactory. The MPEC involve at each time-ste seems comutationnaly har to solve, an nees further investigations. Some other ossible solutions are currently uner reasearch, as changing the mixe comlementarity conitions involve, or using ifferent algorithms for this maximisation. ACKNOWLEDGEMENTS This work has been artially suorte by the LabEx PERSYVAL-Lab ANR--LABX-25-) Ajoint t) : Analytical solution Numerical solution Exlicit) Numerical solution Imlicit) Fig. 3. Numerical solution: inirect aroach - a, b.5,, e 2, f 3, x), N 6 REFERENCES Acary, V., Bonnefon, O., an Brogliato, B. 2). Nonsmooth Moeling an Simulation for Switche Circuits, volume 69. Sringer Science & Business Meia. Brogliato, B. 23). Some ersectives on the analysis an control of comlementarity systems. IEEE Transactions on Automatic Control, 486), Çamlibel, M. an Schumacher, J. 2). On the Zeno behavior of linear comlementarity systems. Proceeings of the 4th IEEE Conference on Decision an Control,

9 , Çamlibel, M.K. 27). Poov Belevitch Hautus tye controllability tests for linear comlementarity systems. Systems & Control Letters, 565), Çamlibel, M.K., Pang, J.S., an Shen, J. 26). Conewise linear systems: non-zenoness an observability. SIAM Journal on Control an Otimization, 455), Cesari, L. 22). Otimization-Theory an Alications: Problems with Orinary Differential Equations, volume 7. Sringer Science & Business Meia. Clarke, F. an De Pinho, M. 2). Otimal control roblems with mixe constraints. SIAM Journal on Control an Otimization, 487), Clarke, F.H. 976). The maximum rincile uner minimal hyotheses. SIAM Journal on Control an Otimization, 46), Georgescu, C., Brogliato, B., an Acary, V. 22). Switching, relay an comlementarity systems: A tutorial on their well-oseness an relationshis. Physica D: Nonlinear Phenomena, 2422), Heemels, W.M.H., Camlibel, M.K., Schumacher, J.M., an Brogliato, B. 2). Observer-base control of linear comlementarity systems. International Journal of Robust an Nonlinear Control, 2), Pang, J.S. an Shen, J. 27). Strongly regular ifferential variational systems. IEEE Transactions on Automatic Control, 522), Passenberg, B., Caines, P.E., Leibol, M., Stursberg, O., an Buss, M. 23). Otimal control for hybri systems with artitione state sace. IEEE Transactions on Automatic Control, 588), Rao, A.V. 29). A survey of numerical methos for otimal control. Avances in the Astronautical Sciences, 35), Shen, J. 24). Robust non-zenoness of iecewise affine systems with alications to linear comlementarity systems. SIAM Journal on Otimization, 244),