PARALLEL TRANSPORT AND MULTI-TEMPORAL CONTROLLABILITY

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1 U.P.B. Sci. Bull., Series A, Vol. 77, Iss. 2, 25 ISSN PARALLEL TRANSPORT AND MULTI-TEMPORAL CONTROLLABILITY Cristian Ghiu, Constantin Udrişte 2 Our idea was to make a bridge between the parallel transport in differential geometry and the controllability of evolutionary linear PDE systems. The purpose is threefold: i to highlight the origin of some linear homogeneous PDE systems; ii to formulate new controllability theorems regarding the bilinear systems; iii to prove that the problem of the Riemannian metric can be thought of as a problem of multitime controllability. The results are concise and confirmed by examples and counterexamples. In particular, the control of metric phases in Riemannian geometry is a new subject in our research group. Keywords: multitime controllability, parallel transport, controls in differential geometry. MSC2: 9B 5, 5C 5, 5B 7.. Controlled multitime linear homogeneous PDE systems.. Multitime linear homogeneous PDE systems Our aim is to study homogeneous PDE systems of the type x t α t = U αtxt, α =, m, where t = t,..., t m R m, x: R m R n = M n, R, and U α : R m M n R are C matrix functions indexed after α =, m. Proposition.. The PDE system is completely integrable iff, for any α, β =, m and any t R m, we have U α t β t + U αtu β t = U β t α t + U βtu α t. 2 Specification. A matrix-valued function F t M n R, t I is said to be i proper on the interval I if F t = ft, A, t I, and a fixed constant matrix A M n R, where f is a scalar function, and ii F t is said to be semiproper on I if F tf τ = F τf t, t, τ I. Lecturer, Department of Mathematical Methods and Models, Faculty of Applied Sciences, University Politehnica of Bucharest, Romania, crisghiu@yahoo.com 2 Professor, Department of Mathematics and Informatics, Faculty of Applied Sciences, University Politehnica of Bucharest, Romania 79

2 8 Cristian Ghiu, Constantin Udrişte The PDE system is called semiproper if U α tu β τ = U β τu α t, t, τ R m, α, β, m functional commutativity of the matrices U α. A semiproper PDE system has a closed form fundamental matrix χt, t = exp U α s ds α. γ t t Consequently, the problem of solving a semiproper system amounts to that of finding a finite form expression for the exponential matrix. If the complete integrability conditions are taken simultaneously with semiproper conditions, then the matrices U α are the components of gradient of a matrix U. Using Theorem 5.5 and Proposition 5.2 of the paper [5], we obtain Proposition.2. Let t R m and x R n. If the matrix functions U α verify the relations 2, then the Cauchy problem {, xt = x } has a unique solution x : R m R n, xt = χt, t x, t R m, where χ, : R m R m M n R is the fundamental matrix associated to the PDE system. The solution x is a function of class C 2. A source for PDE systems of type is the parallel transport in differential geometry on a manifold or on jet bundle of order one, modulo the notations of coordinates and functions. For example, on an n-dimensional manifold M, with local coordinates x = x i, we define a covariant derivative operator, and its components Γ i jk x connection coefficients, to perform the components of covariant derivative, in a way independent of coordinates. i A vector field X = X i on M is called parallel if j X i x = Xi x j x + Γi jk xxk x =, i, j, k, n. ii A second order tensor g = g ij on M is called parallel if k g ij x = g ij x k x Γh ki xg hjx Γ h kj xg hix = g ij x k x Γ h ki xδl j + Γ h kj xδl i g hl x =, i, j, h, k, l, n. For identifying with the PDE system, we need to identify: i the ranges, m,, n; ii t with x; and iii X i, resp. g ij with the unknowns x i. Remark.. In differential geometry, the parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection a covariant derivative or connection on the tangent bundle, then this connection allows one to transport tensors of the manifold along curves so that they stay parallel with respect to the connection. The papers [] [4], [6] [], [5] [7] include information and related techniques, the relevant sources for original ideas in this article.

3 Parallel transport and multi-temporal controllability 8.2. Controlled multitime linear homogeneous PDE systems Linear PDE systems on Lie groups are a natural generalization of linear PDE systems on Euclidean spaces. Indeed, a homogeneous linear system on a Lie group is a controlled system x t α t = uβ αtx β xt, where X α x are linear vector fields. Of course, these PDE systems can be written in the form and conversely. We reconsider the PDE system as a controlled bilinear system, where the matrix functions U α α=,m, U α : R m M n R, are controls of class C, α =, m, which verify the relations 2 on R m, for any α, β =, m. If in addition, for any α, the functions U α are constant, then U α α=,m is called constant control. Definition.. The family of matrix functions U α α=,m, of class C, which verify the relations 2 on R m, is called control for the PDE system. The set of controls, resp. the set of constant controls, is denoted by { } U := U α α=,m Uα α=,m is control for the PDE system, U 2 := { } U α α=,m Uα α=,m is constant control for the PDE system. Hence, the system is completely integrable if and only if U α α=,m is a control function. It is obvious that U and U 2 are real vector spaces and U 2 U. Definition.2. Let t, x, t, x R m R n be two phases. We say that the phase t, x is transferred to phase t, x, if there exists a control U α α=,m, in the PDE system, such that the solution x of the Cauchy problem {, xt = x }, verifies also the relations xt = x. In other words, for the same control U α α=,m, the Cauchy problems {, xt = x } and {, xt = x } have the same solution. We will say that the control U α α=,m transfers the phase t, x into the phase t, x. Remark.2. Let t, x, t, x R m R n. The control U α α=,m U, transfers the phase t, x into the phase t, x, iff χt, t x = x here χ, is the fundamental matrix of the PDE system in which the family U α α=,m is as above, which makes the transfer of phases. This follows immediately from Proposition.2. Remark.. Let x be a solution of the PDE system. If there exists s R m such that xs =, then xt =, t R m. Suppose that the phase t, x transfers to the phase t, x. Then x = iff x =. It is noticed immediately that any control transfers the phase t, into the phase t, t, t R m.

4 82 Cristian Ghiu, Constantin Udrişte For x and x, we further study the transfer of the phase t, x into the phase t, x. In the case n 2, we prove that for any t, t, different, the phase t, x transfers in the phase t, x. a b Lemma.. If a, b R and A =, then e A = e a cos b sin b. b a sin b cos b Proof. For an original proof, we show that e τa = e aτ cosτb sinτb, τ sinτb cosτb R; then we set τ = and we obtain the conclusion. We denote F τ := e aτ cosτb sinτb, τ R. It suffices to show sinτb cosτb that F τ = AF τ, τ R, and F = I 2. The equality F = I 2 is obvious. By computation, we find F τ = ae aτ cosτb sinτb + e aτ b sinτb b cosτb. sinτb cosτb b cosτb b sinτb AF τ = ai 2 + b e aτ cosτb sinτb sinτb cosτb = ae aτ cosτb sinτb + b e aτ sinτb cosτb = F τ. sinτb cosτb cosτb sinτb Lemma.2. Let n. For any x, x R n \ {}, there exists Y, Y R 2 \ {}, and there exists an invertible matrix P M n R, such that Y Y P x = and P x =. 4 O n 2, O n 2, Proof. First, do the proof for the case in which vectors x and x are linearly independent. Let S = Sp {x, x }. The set {x, x } is a basis of the vector subspace S and hence dim S = 2. Then dim S = n 2. Let {c,..., c n } be a basis of the vector subspace S. The set {x, x, c,..., c n } is a basis of R n = M n, R, since S S = R n = M n, R. It follows that the matrix P := c. x x c n is invertible. For j =, n, we have c j x = x c j = c j, x = and c j x = x c j = c j, x =, due to the fact c j S. Hence x 2 x, x P x = x, x and P x = x 2. O n 2, O n 2,

5 Parallel transport and multi-temporal controllability 8 x 2 x, x It is enough to choose Y = and Y = x, x x 2. The vectors Y and Y are nonzero, since x = and x =. Let us suppose now that x and x are linearly dependent. There exists λ R, λ, such that x = λx. Let S = Sp {x }. The set {x } is a basis of S and hence dim S =. Then dim S = n 2. Let {c 2,..., c n } be a basis of S. The set {x, c 2,..., c n } is a basis of c R n = M n, R. It follows that the matrix P = 2. x c n is invertible. For j = 2, n, we have c j x = x c j = c j, x =, since c j S. Consequently x 2 λ x 2 x 2 P x = and P x = λp x =. We choose Y = O n, O n, and Y = λy. The vectors Y and Y are non-zero, since x and λ. Proposition.. Let n 2. For any x, x R n \ {}, there exists A M n R such that e A x = x. If x = x, we can choose A =. If x x, then A can be chosen with rank A = 2. Proof. First, we refer to the case n = 2. Let x = x, x2, x = x, x2. We consider the complex numbers z := x 2 + ix and z := x 2 + ix. The numbers z and z are non-zero since x and x are non-zero. Write z and z in trigonometric form. There exists r, r, and there exists θ, θ [, 2π, such that z = r cos θ +i sin θ and z = r cos θ +i sin θ. Let z = z. Obviously z = r cosθ θ + i sinθ θ. z { r } a b Let G := M b a 2 R a, b R. We consider the function a b f : C G, fa + ib = a, b R. b a It is easily proved that G, with operations of addition and multiplication of matrices, is a field and the function f is an isomorphism of fields. Consequently we have zz = z = fzfz = fz, i.e., r r it follows cosθ θ sinθ θ sinθ θ cosθ θ r Let a := ln x 2 x x x 2 a b, b := θ θ and A := r b a e A = e a cos b sin b sin b cos b = x 2 x x x We use Lemma. and = r cosθ θ sinθ θ r sinθ θ cosθ θ. 6

6 84 Cristian Ghiu, Constantin Udrişte From 5 and 6 we deduce the equality e A x 2 x x x 2 = x 2 x x x 2 Equalizing the second column in the left hand side with second column from the right, and we get e A x x x 2 = x 2 e A x = x. The matrix A, chosen above, is void if and only if a = b = r = r and θ = θ z = z x = x. Hence A = iff x = x. It follows that if x x, then A is non-zero, hence a or b, and from deta = a 2 + b 2 we obtain rank A = 2. We remark that in the case n = 2, we can choose A G A = O 2 or rank A = 2. So we have proved the Proposition for n = 2. Let n. If x = x, we choose A = and it is obvious that e A x = I 2 x = x. Now treat the case where x x. According Lemma.2, there exists Y, Y R 2 \ {}, and there exists an invertible matrix P M n R, such that the equalities 4 hold. We apply the case n = 2 to the vectors Y, Y. We saw that one can choose A G, such that e A Y = Y. A O Let A 2 := 2,n 2 O n 2,2 O n 2,n 2 O n 2,2.. Using the relations 4, we obtain e A 2 e A O P x = 2,n 2 Y e A Y = O n 2, = I n 2 Y O n 2, O n 2, = P x. Hence e A 2 P x = P x. Since P is invertible, it follows P e A 2 P x = x. We choose A = P A 2 P. We get e A x = e P A 2 P x = P e A 2 P x = x. Obviously, rank A= rank A 2 = rank A. If we had rank A 2, then deta =. Since A G and deta =, it follows that A = O 2, hence also A 2 = O n. From A = P A 2 P, we obtain A = O n, and from e A x = x, it follows x = x, which is false. Hence rank A = 2. It follows rank A = 2. In general, the matrix A, appearing in the conclusion of Proposition. can not have the rank. We show this in the following example. Example.. Let n 2, x R n, x and x = ξx, with ξ,. If the matrix A M n R verifies the relations e A x = x, then rank A.

7 Parallel transport and multi-temporal controllability 85 Let us suppose that rank A =. Let J M n C, be the canonical Jordan form o the matrix A. Rank J = rank A =. If on the principal diagonal of J should exists at least two nonzero elements, then rank J 2, which is impossible. It follows that at least n from the eigenvalues of the matrix A are zero, and the last eigenvalue, λ, is real, since λ = T ra R. Then the matrix e A has n eigenvalues equal to, an the n-th eigenvalue is e λ. It follows that e A + ξi n has n eigenvalues equal to + ξ, and the n-th eigenvalue is e λ + ξ. Hence dete A + ξi n = + ξ n e λ + ξ >. We deduce that the matrix e A + ξi n is invertible. The equality e A x = x is equivalent to e A x + ξx = or e A + ξi n x =. Since the matrix e A + ξi n is invertible, it follows x =, that is false. We denote by U the set of families of functions U α α=,m, U α : R m R n, with the property U α are all identically zero, or there exists α =, m and there exists a matrix M M n R, with rank M = 2, such that { M, t R U α t = m ; if α = α O n, t R m ; if α α. Note that for such families, the relations 2 are true. It follows that the elements of the set U are constant controls for the PDE system. Hence U U 2 U. Theorem.. Let n 2. For any t, x, t, x R m R n, with x, x and t t, there exists a control in U which transfers the phase t, x into the phase t, x. Proof. Since t t, there exists α =, m, such that t α t α. According the Proposition., there exists A M n R, with A = or rank A = 2, such that e A x = x. Let M = U α t = t α tα { M, t R m ; if α = α O n, t R m ; if α α. A. We choose Obviously, U α α=,m U. Taking into account the Proposition.2 and the formula χt, t = e Mαtα t α, t, t R m R m, 7 it follows that the solution of the Cauchy problem {, xt = x } is x : R m R n, xt = e tα t α M x, t = t,..., t m R m. We obtain xt = e tα tα M x = e A x = x, therefore chosen control transfers the phase t, x into the phase t, x.

8 86 Cristian Ghiu, Constantin Udrişte The Theorem. says that for n 2, in the case of the PDE system, it is sufficient to consider as controls only elements of the set U. In the case n =, we have the next result Theorem.2. Let n =, t, x, t, x R m R, with x and x. a If the phase t, x transfers into the phase t, x, then sgnx = sgnx. b Let us suppose that sgnx = sgnx and t t. Let α =, m, such that t α t α. We consider the functions U α : R m R, U α t = t α tα ln x x, t R m ; if α = α, t R m ; if α α. Then U α α=,m is a constant control for the PDE system, and transfers the phase t, x into the phase t, x. Proof. a The function χ, t : R m R is continuous. Since for any t R m, the matrix χt, t is invertible, i.e., χt, t, it follows that the function χ, t has constant sign. But χt, t = > ; hence χt, t >, t R m. According the Remark.2, we have χt, t x = x. Since χt, t >, it follows that x and x have the same sign. b We have x >, since sgnx = sgnx. Hence U α is well definite. x Because the relations 2 are obviously true, it follows that U α α=,m U 2. Using the formula 7, we obtain, ln χt, t x = e x x x = x x x = x. Consequently the control U α α=,m transfers the phase t, x into the phase t, x, according the Remark.2. The Theorem.2 says that, for n =, in the case of the PDE system, it is sufficient to consider as controls only constant functions form 8. For the PDE systems of the type we may consider other sets of controls, how was it U, for example. Also, it may impose conditions on solutions x ; for example, require that x take values in a given set, included in R n. 2. Metric parallelism metric phase transfer In the paper [4] one solves the following problem: giving a second order differential equation, under what conditions the graphs of solutions of this equations are geodesics for a Riemannian manifold D R 2, g ij? Obviously g = g ij i,j {,2}, is a Riemannian metric that must be determined. It shows that it is necessary that the given equation to be a Riccati second order equation and Γ ij =. In these conditions, the coefficients of the Levi-Civita connection, Γ 2 ij, are expressed with respect to the functions which defines the Riccati equation and conversely. 8

9 Parallel transport and multi-temporal controllability 87 Equivalently, the Riccati equation is known if and only if the coefficients Γ 2 ij, i, j {, 2} are known. Let us show that g, g 2, g 22 is the solution of the linear homogeneous PDE system 9, written below. We consider the linear homogeneous PDE system x t = U tx, x t 2 = U 2tx, 9 where t = t, t 2 R 2, x = x, x 2, x : R 2 R R,, and U, U 2 : R 2 M R are C matrix functions of the form 2Γ 2 t 2Γ 2 U t = Γ 2 2 t 2 t Γ2 t, U 2 t = Γ 2 2Γ 2 22 t Γ2 2 t 2 t 2Γ 2 22 t. Note that systems 9 are a special cases of the PDE systems, with m = 2, n =. In the paper [4] was proved the following result Theorem 2.. We consider given C functions Γ 2, Γ2 2, Γ2 22 : R2 R. a The following statements are equivalent: i On R 2 there are true the relations Γ 2 2 t 2 = Γ2 22 t, Γ 2 2 t + Γ 2 2 Γ 2 2 = t 2 + Γ2 Γ ii There exists the C function W : R 2 R, such that on R 2 we have the relations W t 2 >, Γ2 = 2 W t 2 2, Γ W 2 = t 2 2 W t t 2 2, Γ W 22 = t 2 2 W t 2 2 W. 2 t 2 iii The PDE system 9 has at least a solution x : R 2 R R,, of class C. b Suppose the equivalent statements i, ii, iii are true. Let W be a function as in the point ii. Then all the solutions, x = x, x 2, x : R 2 R R,, of the PDE system 9, are of the form x = a W t + c 2 + b; x 2 = a W W W 2 t + c t 2 ; x = a t 2 with a >, b, c arbitrarily real constants. In these conditions, the functions x, x 2, x are of class C 2. Proposition next check immediately. Proposition 2.. Suppose that we are considering Theorem 2. and that the equivalent statements i, ii, iii are true. Let W as in the statement ii of the Theorem 2..

10 88 Cristian Ghiu, Constantin Udrişte Then t R 2, x R 2,, the Cauchy problem {9, xt = x } has a unique solution. This is given by the formulas, where x a = W 2, b = t 2 t x x x 2 x 2 x = x x2 2 x, 4 c = x 2 x x W t t W t 2 t = x2 x W t 2 t W t t. 5 Definition 2.. The pair of matrix functions U, U 2 is called control for the PDE system 9 if there exists a function W : R 2 R, of class C, with W t >, t2 t R 2, such that, U, U 2 are defined by the relations, for Γ 2 ij given by the formulas 2. The function W defines completely the control. We denote by U W, the control defined the function W. We observe that the pair U, U 2 is control for the PDE system 9 iff the functions Γ 2 ij, which define U and U 2, verify the equivalent statements i, ii, of the Theorem 2.. For the PDE system 9, the phase transfer is similar to the exposed in Section, for the PDE system. It differs only choice controls; they are now the ones presented in Definition 2.. Proposition 2.2. For any control U, U 2 and any solution x of the PDE system 9, there exists a constant k R, such that x t x2 t 2 x t i.e., the function F x, x 2, x = x x x = k, t R 2, 6 is a first integral for the PDE system Proof. Any solution is given by the formulas. We have W 2 W 2 x tx t = a 2 t t + c t 2 t + bx t = x 2 t 2 + bx t. Hence x tx t = x 2 t 2 + bx t, equivalent to x t x2 t 2 k = b. x t Proposition 2.. Let x, x R 2, and t, t R 2. a If the phase t, x transfers to the phase t, x, then x x2 2 x b Let us suppose that the equality 7 is true. = b. We choose = x x2 2 x. 7

11 Parallel transport and multi-temporal controllability 89 Let W : R 2 R, of class C, with W t 2 t >, t R2. Then the control U W transfers the phase t, x into the phase t, x iff x 2 x x W 2 = t 2 t x W 2 and 8 t 2 t W t 2 t W t t = x2 x W t 2 t W t t. 9 Proof. a If the phase t, x transfers to the phase t, x, then there exists a control, such that the Cauchy problems {9, xt = x } and {9, xt = x } have the same solution x. According the Proposition 2.2, there exists k R, such that the relation 6 holds true, t R 2. Hence we have x t x2 t 2 x t = x t x2 t 2 x t = k x x2 2 x = x x2 2 x = k. b The control U W transfers the phase t, x into the phase t, x iff the Cauchy problems {9, xt = x } and {9, xt = x } have the same solution x. This is equivalent, according the Proposition 2., to the fact that the numbers a, b, c are a = x W 2 = t 2 t x W 2, b = x x2 2 t 2 t x = x x2 2 x c = x2 x W t 2 t W t t = x2 x W t 2 t W t t. and Theorem 2.2. Let x, x R 2,, such that the relation 7 is true. Then, for any t, t R 2, t t, the phase t, x transfers to the phase t, x. a If t t, we consider W : R 2 R, W t, t 2 = A t t 2 2 where B = ln x ln x 2t t, A = t t + e Bt t t 2 t 2, t, t 2 R 2, x 2 x2 x x x x Bt 2 t 2. x Then the control U W transfers the phase t, x into the phase t, x. b If t = t, t2 t2, x x, we consider W : R 2 R, W t, t 2 = B t eabt e Bt2 t 2, t, t 2 R 2, where A = x x 2 x2, B = ln x ln x x x x 2t 2 t2. Then the control U W transfers the phase t, x into the phase t, x.

12 9 Cristian Ghiu, Constantin Udrişte c If t = t, t2 t2, x = x, we consider W : R 2 R, W t, t 2 = e At t t 2 t 2, t, t 2 R 2, where A = x2 x2. x Then the control U W transfers the phase t, x into the phase t, x. Proof. One applies the Proposition 2.. We have to show that W t 2 > and that the equalities 8, 9 are true. a W t 2 t, t 2 = e Bt t >, W t t, t 2 = At t + Be Bt t t 2 t 2. W t 2 t =, W t t =, W t 2 t = e Bt t W, t t = At t + Be Bt t t 2 t 2. The equality 8 is equivalent to x = x e 2Bt t ebt t x = B = ln x ln x x 2t t, the last equality being true. The equality 9 is equivalent to x 2 x = x2 x e Bt t At t Be Bt t t 2 t 2. 2 Since e Bt t x =, the equality 2 is equivalent to x x 2 x = x2 x x x At x t B t 2 x t 2, or A = x 2 t x2 x t x x x Bt 2 t 2, x which is true. Analogously, we treat the cases b and c. Acknowledgments The work has been funded by the Sectoral Operational Programme Human Resources Development 27-2 of the Ministry of European Funds through the Financial Agreement POSDRU/59/.5/S/295. Partially supported by University Politehnica of Bucharest, and by Academy of Romanian Scientists.

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