TIME-OPTIMAL CONTROL OF A PARTICLE IN A DIELECTROPHORETIC SYSTEM. Dong Eui Chang Nicolas Petit Pierre Rouchon

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1 TIME-OPTIMAL CONTROL OF A PARTICLE IN A DIELECTROPHORETIC SYSTEM Dong Eui Chang Niolas Petit Pierre Rouhon CAS, Eole des Mines de Paris 60, bd. Saint-Mihel, Paris, CEDEX 06, Frane Abstrat: We stud the time-optimal ontrol of a partile in a dieletrophoreti sstem. This sstem onsists of a time-varing non-uniform eletri field whih ats upon the partile b induing a dipole moment within it. The interation between the dipole and the eletri field generates the motion of the partile. The ontrol is the voltage on the eletrodes whih reates the eletri field. Suh sstems have wide appliations in bio/nanotehnolog. In regard to time-optimal ontrol, we address the issue of existene and uniqueness of optimal trajetories, and expliitl ompute the optimal ontrol and the orresponding minimum time. Kewords: time-optimal ontrol, biotehnolog, nanotehnolog, dieletrophoresis. INTRODUCTION We onsider the motion of a neutrall-buoant and neutrall-harged partile on an invariant line in a hamber filled with fluid flowing at low Renolds number and with a parallel eletrode arra at the bottom of the hamber; see Figure. A fore is reated b the interation between a non-uniform eletri field and the dipole moment indued in the partile. The resultant motion is alled dieletrophoresis (DEP); see (Pohl, 978). We are interested in the time-optimal ontrol of this sstem whih an be desribed, after a nonlinear hange of oordinates, b ẋ = u + αu 2 () ẏ = + u. (2) The state (x, ) R 2 and the single ontrol u satisf x(0) = given, (0) = 0, (3) x(t f ) = given, (t f )= free, (4) u (5) where the parameters α and satisf α<0, > 0. (6) Variable x desribes the displaement of the partile. Variable desribes the exponentialldeaing part of the indued dipole moment. Control u is the ommon voltage on eah eletrode. Parameters α and depend on the permittivities and ondutivities of the partile and the fluid medium. Dieletrophoresis has wide appliations in bio/nanotehnolog, in partiular, in the separation of bio/nano-partiles (Hughes, 2002; Jones, 995). We here address the ase of a single partile to underline ke features of DEP suh as the dipole indution and the boundar ontrol. We hope that this will invite ontrol researhers to the appliation of DEP sstems. 2. OVERVIEW OF MAIN RESULTS Time optimal trajetories satisf the following dnamis: with onditions (3) (5) and ẋ = u + αu 2 (7) ẏ = + u (8) = u (9)

2 (0) = 0 = to be found, (t f ) = 0 (0) where u(t) = arg max H((t),(t),v) v with H(,, v) =αu 2 +( + )u. () See (2) for the optimal ontrol u. When x f <x(0), there are no Lebesgue measurable time-optimal ontrol funtions resulting in x(t f )=x f even though x f is reahable. We now onsider the ase of x f > x 0. Timeoptimal ontrol law exists if and onl if the parameters, α and satisf ( + α) > 0. Let Λ=( 2α +α, /) and Λ =( 2α +α, 2α), Λ 2 =[ 2α, /) (2) where it is understood that the deomposition of ΛintoΛ and Λ 2 is done onl when (+2α) > 0. For the sake of onveniene, let us define three sentenes as follows: P := [( + 2α) > 0] [ 0 Λ ], P2 := [( + 2α) > 0] [ 0 Λ 2 ], Q := [( + 2α) 0] [ 0 Λ] where is the logial onnetive, AND. Let us define a stritl inreasing onto funtion X :Λ (0, ) b equation (22) if P Q, X ( 0 )= equation (24) if P2 where is the logial onnetive, OR. Let us define another funtion T on Λ b equation (28) if P Q, T ( 0 )= equation (29) if P2. Classiall, we all a trajetor of (7) (9) satisfing (3) (5) and (0), an extremal. Let us all an ar of an extremal a basi ar if the projetion of the ar onto the -plane starts from 0 Λ(resp. 0 ( Λ)) and ends on Λ 0(resp.( Λ) 0), going through the first (resp. third) quadrant of the plane. We all an extremal an n-shot extremal with n N if the maximum number of basi ars in the extremal is n. To deompose extremals into finite ars when [P Q], let us introdue some notations. An ar with the linear ontrol u =( + )/( 2α) on a time interval of length Δt AB ( 0 ) in (25), is denoted b γ 0 L.Letγ0 + (resp. γ 0 ) denote an ar with u = (resp. u = ) in a time interval of length Δt BC ( 0 ) in (27). The onatenation Γ 0 ± = γ 0 L γ0 ± γ 0 L is alled a basi ar where the onatenation is suh that the leftmost one omes first and the rightmost one omes last and the onatenation urve is alwas a ontinuous urve. An ar with the linear ontrol u =( + )/( 2α) is alled an idling ar if its projetion (, ) starts from the positive (resp., negative) -axis, goes through the fourth (resp., seond) quadrant in the -plane, and finall ends on the negative (resp., positive) -axis. An idling ar is denoted b γ idling ; see (b) of Figure 3. Its duration is T idling in (30). Hene, when [P Q], we an express n-shot extremals as Γ 0,n ±, n N whih is defined as follows: Γ 0, ± ; ± =Γ 0 Γ 0,k ± = for k 2 with Γ 0,k ± γ idling Γ 0 Γ 0,k ± γ idling Γ 0 (0) = ± 0. if k is even, ± if k is odd Let us now talk about the existene and uniqueness of optimal trajetories. If (+2α) > 0, there exist exatl two time-optimal trajetories for a given x f >x(0), and the are basi ars. Here is the proedure of onstruting them:. Find 0 = X (x f x(0)) Λ, 2. Set (0) = ± 0, 3. The minimum time ost is T ( 0 ) and the optimal trajetories γ opt are Γ 0 γ opt = ± if P, basi ar with u = ± ifp2. If ( + 2α) 0, we do not have an general proof of the uniqueness of optimal ontrol. However, we have a finite proedure of finding all optimal ontrol laws for a given x f >x(0) as follows:. Define two sequenes, for k N, ( ) 0,k = X xf x(0), k T k = kt ( 0,k )+(k )T idling 2. Find n = arg min k T k : k N} (it alwas exists and is less than + (T /T idling )), 3. Set (0) = ± 0,n, 4. The minimum time ost is T n and the orresponding optimal trajetories are Γ 0,n ±. If there are j integers in step 2 to give the minimum time, then there are exatl 2j time-optimal trajetories. We remark that eah basi ar in the k-shot extremal Γ 0,k,k ± equall ontributes (x f x(0))/k to the inrement of x and the idling ars in between make no ontributions. 3. DERIVATION OF DYNAMICS We briefl derive the dnamis in () and (2), and explain the onditions on the state and parameters in (3) (6); see (Chang et al., 2003) for

3 more detail. Consider a neutrall harged partile in a hamber with a fluid medium and a parallel eletrode arra at the bottom as in (a) of Figure where d is the width of eah eletrode, and d 2 is the length of the gap between two eletrodes. As the eletrodes are long enough ompared with the size of partiles, we ma assume that there are infinite number of infinitel long eletrodes. Due to this smmetr, we will onsider the motion of the partile in the vertial plane as in (b) of Figure. Let (q, p) R 2 be the oordinates in (b) d d 2 p q hek that the dieletrophoreti fore on the p axis is along this axis. This vertial dieletrophoreti fore on the p axis is denoted b F dep (p, t). It is of the form F dep (p, t) =F (p)u(t)(g u)(t) with F (p) 0onp 0, lim p F (p) = 0, and F (p) = 0 onl at p = 0; see (Chang et al., 2003). Let us assume that the partile is neutrall buoant and the medium fluid flows at low Renolds number. Thus, the gravitational fore and the buoant fore anel and the inertial term m p is ignorable. The onl fores on the partile is the drag and the DEP fore. Hene, the motion of the partile on the p axis an be desribed b (a) (b) f ṗ + F (p)u(t)(g u)(t) =0. (5) Fig.. The dieletrophoreti sstem is a hamber filled with a fluid medium where there is a parallel arra of eletrodes at the bottom. of Figure. We impose the ommon boundar voltage on eah eletrode as follows: V db (t) =V 0 u(t), u(t). The eletri field vetor E(q, p, t) R 2 in p 0} is given b E(q, p, t) = V (q, p, t). This eletri field indues a dipole moment, m, on a single-laered spherial partile as follows: m(q, p, t) =g(t) E(q, p, t) where denotes time-onvolution and the Laplae transform G(s) of the (transfer) funtion g(t) is given b G(s) =a + b s + where a =4πr 3 ɛ m (ɛ p ɛ m )/(ɛ p +2ɛ m ), ( σp σ m b = a σ ) p +2σ m, (3) ɛ p ɛ m ɛ p +2ɛ m =(σ p +2σ m )/(ɛ p +2ɛ m ) (4) where r is the radius of the partile, ɛ p (resp., ɛ m ) is the permittivit of the partile (resp., medium) and σ p (resp., σ m ) is the ondutivit of the partile (resp., medium). The interation between the eletri field and the indued dipole moment reates a fore F dep. It is alled dieletrophoreti fore and given b F dep (q, p, t) =(m(q, p, t) )E(q, p, t). As the vertial motion of partiles in the whole hamber an be pratiall represented b that of partiles on the p axis, we are interested in the motion of partiles on the p axis. One an where f>0isthe drag onstant. We assume that b in (3) is nonzero, whih generiall holds. Then equations () and (2) ome from (5) where x and are defined b x = p ɛ f dz; Y (s) = U(s) (6) bf(z) s + for p ɛ where ɛ is a positive number and Y (s) and U(s) are the Laplae transforms of (t) and u(t), and α is defined b α = a/b. If a partile is lose to the eletrode, then additional phsial/hemial fores other than the DEP fore start to appear in the dnamis (Hughes, 2002; Pohl, 978), so the parameter ɛ in (6) defines the region where the dnamis (5) is valid. Phsiall, is the exponentiall indued part of the dipole moment, so we have the initial ondition (0) = 0. As we are not interested in the final state of the indued dipole, we have (t f ) = free. Depending on the sign of b, the original region p ɛ} is mapped to x 0} or x 0}. However, in this paper, we ignore the state onstraint on x, allowing for x on the whole real line. In a future publiation, we will address the optimization problem with the state onstraint on x. We also make the following assumption on the signs of parameters α and α<0, > 0. Assumption >0 is imposed b phsis; see (4). However, ondition α<0 is for onveniene. The ase of α 0 an be treated in the similar manner.

4 4. MAIN DISCUSSION 4. Non-existene of optimal ontrol for x f < x(0). We will show that there are no time-optimal (Lebesque) measurable ontrols for x f < x(0) even though x f is reahable. For a T>0, let us define a sequene of funtions u T n :[0,T] ±} as follows: u T n (t) = sign(sin((2πt)/(nt ))), n N. (7) Lemma. Let f be a ontinuous funtion on [0,T]. Then, lim n t 0 f ut n = 0, uniforml in t [0,T]. The substitution of u from (2) to () ields ẋ = ẏ αu 2. For an T>0wehave x(t ) x(0) = T T 2 (T ) α u 2 αt (8) 0 0 where the last inequalit holds b (5). This implies that in the time interval [0,T], the negativel farthest reahable point is at best x(0) + αt.let (x n, n ) be the solution to () and (2) on [0,T] with ontrol u T n in (7). B Lemma, (8), and the definition of u T n, one an show lim (x n(t ) x(0)) = αt. n We have onstruted a sequene u T n } of ontrol laws suh that the orresponding x n (T )} onverges to the infimum of the reahable points of x. However, the sequene of funtions u T n } does not onverge to a measurable funtion. One an prove that for x f <x(0), the infimum of the reahable time is T = (x(0) x f )/( α), but there are no time-optimal (Lebesgue) measurable ontrols to reah x f in time T. We remark that for x f = x(0), ontrol u = 0 with t f = 0 is triviall the timeoptimal ontrol. Hene, in the rest of the paper, we onl onsider the ase of x f >x(0). 4.2 Pontragin maximum priniple, Neessar ondition on parameters and Disrete smmetr. We derive, from the Pontragin Maximum Priniple (PMP), neessar onditions for time-optimal trajetories. Let us define the PMP Hamiltonian, H, for the time-optimal ontrol as follows (Pontragin et al., 962): H(x,, x,,u)= x αu 2 +( x + )u where ( x, ) is the vetor dual to (x, ). Let H (x,, x, ) = max H(x,, x,,u). u Consider sstem (), (2) with onditions (3) (5) and (6). Let u(t) be a time-optimal ontrol and (x(t),(t)) be the orresponding trajetor. Then, b the PMP, it is neessar that there exists a ontinuous vetor ( x (t), (t)), whih is not identiall zero, suh that ẋ = u + αu 2 ; ẏ = + u, x =0; = x u. (9) Additionall, the following must be satisfied:. u(t) = arg max x(t), (t),v), v 2. H (t) = onstant 0, 3. (t f ) = 0 (transversalit ondition). (20) As x = 0 in (9), x (t) is onstant in time. We onsider the three different ases; x =0, x < 0 and x > 0. A simple integration of (9) with the transversalit ondition (20) implies that x = 0 implies (t) 0. Hene, there annot be an optimal trajetories with x = 0 beause the vetor ( x, ) annot be identiall zero b the PMP. In the ase of x < 0, one an show, b simple omputation, that no extremals satisf both the initial ondition (0) = 0 and the transversalit ondition (20). Hene, there exist no optimal trajetories with x =0. We now assume that x > 0. Let = / x ; H(,, u) =H/x. Then equation (9) is replaed b (9), giving us the set of equations (7) (9). As α<0, the ontrol u maximizing H (or equivalentl H) is given b + if + > 2α, + u = if + 2α, (2) 2α if + < 2α. The (, ) dnamis with the ontrol (2) have three qualitativel different phase portraits depending on the sign of ( + α). For example, the origin (, ) =(0, 0) beomes a saddle, a degenerate equilibrium, and a enter, respetivel if ( + α) is negative, zero and positive, respetivel. A simple but thorough phase portrait analsis shows we omit the detail that when ( + α) < 0or ( + α) = 0, there are no trajetories satisfing the initial ondition (0) = 0 and the final ondition (t f ) = 0 (equivalentl, (t f ) = 0). Hene, time-optimal trajetories exist onl when +α > 0 whih is assumed in the following of the paper. In this ase, the dnamis of (, ) have two qualitativel different phase portraits depending on ( + 2α) 0, or ( + 2α) > 0.

5 The portraits are given in Figure 2. A big differene between the two is whether the intersetion of the positive -axis and the swithing line + = 2α is above or below /. u = + 2α u = starting from the axis and ending at the axis an be deomposed into parts, eah of whih is invariant under S or S 2. This implies Δx = 0. However, we alread know what the timeoptimal ontrol is for Δx =0.Itisu = 0 with t f = 0. Therefore, all time-optimal trajetories are ontained in the shaded region beause onl the extremals in the shaded region an satisf (0) = 0 and (t f )=0. u = + = 2α + =2α (a) ( + 2α) 0 S 2 (A) S 2 (B) S 3 (B) A B S (B) S (A) S 3 (A) u = (a) Smmetries (b) Idling ars u = + 2α Fig. 3. (a) Disrete smmetries in the linear region. (b) Idling ars. u = + = 2α + =2α Let us onsider smmetr S 3.LetZ be the vetor field in (7) (9) with the ontrol u in (2). Then we have S 3 Z = Z S 3, whih implies Δt AB =Δt S3(A)S 3(B) and Δx AB =Δx S3(A)S 3(B) in (a) of Figure 3. (b) ( + 2α) > 0 Fig. 2. The phase portrait in the -plane when x > 0 and ( + α) > 0. Depending on the sign if ( + 2α), point (0, 2α) isaboveor below point (0, /) on the axis. We now disuss about the disrete smmetr in the phase portraits. Let us define three refletions, S i,i=, 2, 3 as follows: S (x,, ) =(x,, ), S 2 (x,, ) =(x,, ), S 3 (x,, ) =S S 2 (x,, ) =(x,, ). Denoting the dnamis in (7) (9) b Z L with the linear ontrol u =( + )/( 2α), one an hek S i Z L = Z L S i, i =, 2. The linear vetor field, Z L, is invariant under refletions, S and S 2, up to the time-reversal. In (a) of Figure 3 the duration Δt AB from A to B along the trajetor with u =( + )/( 2α) in the (, )-plane equals Δt Si(B)S i(a), i =, 2. Also, the orresponding inrements in x satisf Δx AB = Δx Si(B)S i(a), i =, 2. This implies that in the white region surrounded b the shaded region in Figure 2 there annot be an optimal trajetories beause an orbits 4.3 Constrution of X, T and T idling. Let us all the part of the shaded region in Figure 2 ling in the first quadrant a basi region. We inlude the part on the -axis and the -axis but we exlude the boundar of the shaded region whih is stritl inside the first quadrant. Let us denote the open interval on the axis of the basi region b Λ. It is given b Λ= R + M( α, α) <M(0,) <M( ( + 2α)/, /)} =( 2α +α, 2α ( + α)/( α)). where M is the Hamiltonian H in () with u = ( + )/( 2α). One an see that the ar of an extremal ontained in the basi region is a basi ar, whih is defined in 2. Let us onstrut a map X on Λ, whih measures the inrement of x along a basi ar starting with (0) Λ. We show the onstrution in the ase of ( + 2α) > 0 onl, as the other ase an be done similarl. When ( + 2α) > 0, we an deompose ΛintoΛ and Λ 2 as in (2). The basi ar with (0) Λ is exatl Γ (0) + defined in (2), whih orresponds to ABCD in Figure 4. The basi ar with (0) Λ 2 orresponds to FG in Figure 4. Suppose (0) = 0 Λ.BtheS -smmetr, the x-inrement along ar AB and along ar CD

6 F 2α A B D C G (, ) Fig. 4. Constrution of the x-inrement map X when ( + 2α) > 0. anel, so X ( 0 )isthex-inrement along ar BC, whih an be omputed as follows: X ( 0 )= [ 2α 2 C( 0 ) ( + α + ) ( )] +2α + C ( 0 ) ln (22) C ( 0 ) where B and C are the -oordinates of points B and C and are given b B ( 0 )= α α 2 +( 2 0 4α2 )/( 4α), C ( 0 )= B ( 0 ) 2α. (23) In the similar wa, one an ompute X ( 0 ) with 0 Λ 2, whih is given b X ( 0 )= [ 0 ( + α)ln( ] 0). (24) Regarding X in (22) as a funtion of C,one an hek dx /d C > 0onΛ.As C in (23) is a stritl inreasing funtion of 0, it follows that X is stritl inreasing on Λ. B omputing dx /d 0 > 0, one an show that X in (24) is also stritl inreasing on Λ 2.Thus,X is a stritl inreasing funtion on Λ. One an also show X (Λ) = (0, ). Let T be the time duration of basi ars with (0) Λ, so it beomes a funtion on Λ. We onl show its onstrution in the ase of ( + 2α) > 0 for the spae limit. Consider a basi ar with 0 Λ.ItisofformΓ 0 +, e.g., ar ABCD in Figure 4. B the S -smmetr, the duration of AB equals that of CD, whih is denoted b Δt AB and given b Δt AB ( 0 )= ω sin ( 2αω B ( 0 )/ 0 ) (25) where ω = ( + α)/( α) > 0. (26) A simple integration ields the duration, Δt BC of the ar BC, whih is given b Δt BC ( 0 )= ( ln ) B( 0 ). (27) C( 0 ) Hene, T on Λ is given b T =2Δt AB +Δt BC. (28) B a diret integration in the similar wa, one an show that T on Λ 2 is given b T ( 0 )=( /) ln( 0 ). (29) Reall the definitions of idling ars and the idling time in 2; see also (b) of Figure 3. The name of idling ars omes from the fat that the do not ontribute an inrements in x b the S -and S 2 -smmetr. One an ompute the idling time, T idling, as follows: T idling =(/ω) sin ( 2αω) (30) with ω in (26). Notie that the idling time is independent of the oordinates of the initial points on the -axis. 4.4 Disussion on the existene and uniqueness of optimal trajetories. We an lassif extremals with k N defined in 2. Notie that for k 3, the (, )-projetion urves of all the k-shot extremals are losed urves in the -plane b the S i,i=, 2, 3 smmetr. Given x f > x(0), one an show that for eah n N there exist exatl two n-shot extremals one is the refletion image of the other b the S 3 smmetr suh that both produe x f. In the ase of ( + 2α) > 0, we an prove that the two one-shot extremals are the time-optimal ones. The wa of onstruting them is given in 2. However, in the ase of ( + 2α) 0, we have not found an uniqueness proofs so far. Instead, we provide a finite algorithm of finding timeoptimal trajetories in 2. With this algorithm, for example, we perform a simulation with the following speifiation: α = 3/4, =,x(0) =, and x f = 2. In this ase, two one-shot extremals are the optimal trajetories and the minimum time is t f = The optimal ontrol u(t) and the optimal trajetor (x(t),(t)) with 0 = are given in Figure (t) u(t) t x(t) Fig. 5. Optimal solution for a sample ase. Aknowledgements. We would like to thank Bernard Bonnard, Jean Lévine, Jerr Marsden, Laurent Pral and Banavara Shashikanth for several helpful suggestions.

7 REFERENCES Chang, D. E., S. Loire and I. Mezić (2003). Closed-form solutions in the eletrial field analsis for dieletrophoreti and travelling wave inter-digitated eletrode arras. J.Phs. D:Appl. Phs. 36(23), Hughes, M. P. (2002). Nanoeletromehanis in Engineering and Biolog. CRC Press. Boa Raton. Jones, T. B. (995). Eletromehanis of Partiles. Cambridge Universit Press. New York. Pohl, H. A. (978). Dieletrophoresis. Cambridge Universit Press. Cambridge, UK. Pontragin, L. S., V. G. Boltanskii, R. V. Gamkrelidze and E. F. Mishhenko (962). The Mathematial Theor of Optimal Proesses. John Wile & Sons, In.. New York.