Dynamical canonical systems and their explicit solutions

Dynamical canonical systems and their connections with the classical (spectral) canonical systems are considered. We construct B\"acklund-Darboux transformation and explicit solutions of the dynamical canonical systems. We study also those properties of the solutions, which are of interest in evolution and control theories.


Introduction
In this paper we consider the system where I p stands for the p × p idenity matrix, m 1 ≥ 0, m 2 ≥ 0, and we put m := m 1 +m 2 > 0, that is, j is an m×m matrix and H(x) is an m×m matrix function. The case m 1 > 0 and m 2 = 0 includes, for instance, the transport equation and the case m 1 = m 2 includes the equation of a vibrating string (see, e.g., [16]). We assume that x takes values on a semiaxis and without loss of generality consider the semiaxis x ≥ 0.
In the case m 1 = m 2 = p, we will show (using formal Fourier transformation) an interesting correspondence between the system (1.1) and the well-known canonical system d dx y(x, λ) = iλJH(x)y(x, λ), H(x) ≥ 0, J = 0 I p I p 0 , (1.2) where λ ∈ C and C stands for the complex plain. Similar to the corresponding terminology for Dirac systems [2], we call system (1.1) the dynamical canonical system and system (1.5) -the spectral canonical system. Since spectral Dirac (or Dirac-type) systems are equivalent to a subclass of spectral canonical systems (see, e.g., [32, Sect. 1.1]), dynamical Dirac systems are equivalent to a subclass of dynamical canonical systems. (On the applications of the dynamical Dirac systems see [2,4,23] and further references therein.) Using Bäcklund-Darboux transformation for the spectral canonical system [25] we construct Bäcklund-Darboux transformation and explicit solutions for the dynamical canonical system (1.1). (See Theorem 2.1 and Corollary 2.3, respectively.) We also study asymptotic behavior of these explicit solutions (see Theorem 3.4).
It is important that if jH 0 (x) (where H 0 (x) is the Hamiltonian of some initial dynamical system) is linear similar to some diagonal matrix D(x), then the transformed Hamiltonian H(x) has the same property, that is, jH(x) is similar to the same matrix D(x). We note that the linear similarity of jH to some diagonal matrix function D is an essential assumption in the main result (Theorem 1.5) in [16] on the operators Af (x) = j d dx H(x)f (x) generating a C 0 -semigroup for (1.1). In fact, system (4.2) (which is more general than (1.1)) is dealt with in [16].
Each generalized Bäcklund-Darboux transformation, in our GBDT version, is determined by some initial system and by some triple {A, S(0), Π(0)} of parameter matrices such that We show that the energy of the constructed solutions is directly expressed via A and S(0) as well (see (3.5)).
Further in the Introduction we discuss the mentioned above topics in greater detail and give various references. We note that canonical (spectral canonical) systems have been actively studied in the literature (see the books [7,14,32,34], quite recent papers [1,21,36] and various references therein). Using circumflex accent to denote Fourier transform in the complex domain, that is, setting (1.4) and formally applying Fourier transform to both sides of (1.2), we derive When the values of the matrix function (of Hamiltonian) H(x) are strictly positive definite (i.e. H(x) > 0), we set Y (x, t) = H(x) y(x, t) and rewrite the last equation in an equivalent form One may consider a more general case of (1.2), where J is an arbitrary matrix such that J = J * = J −1 and J * denotes the matrix adjoint to J. Then the same formal Fourier transform of (1.2) leads to system (1.5) with the same (more general) J. Since each matrix J, such that J = J * = J −1 , is unitarily equivalent to some j given in (1.1), system (1.5) is equivalent to the system (1.1) with the corresponding j.
Using GBDT for (1.2) and Fourier transformation (1.4), we could obtain explicit solutions of the dynamical system (1.5). For example, in a related case of spectral and dynamical Dirac systems we established [31] simple rigorous connections between Weyl functions of the spectral Dirac systems and response functions of the dynamical Dirac systems and used the procedure of solving inverse problem for the spectral Dirac system [27,30] in order to recover the dynamical Dirac system from its response function. However, in the case of canonical systems we use GBDT for the system (1.2) for heuristic purposes only. In spite of similarity of many formulas in the dynamical and spectral cases, it seems simpler (as soon as one knows the result) to construct GBDT and explicit solutions of the system (1.1) directly.
The condition that A generates a C 0 -semigroup (see [16,Theorem 1.5]) which was discussed above, is closely connected with the well-posedness of system (1.1) with boundary control [37]. We note that the study of wellposedness is basic in many evolution and control problems and a large number of recent publications is dedicated to this topic (see, e.g., [3,5,6,9,18,22,35]).
GBDT and explicit solutions for the dynamical canonical system are constructed in Section 2. In Section 3 we study asymptotic behavior as well as energy and some other properties of the obtained solutions. We show also that jH in our case is linear similar to simple diagonal matrices. The last section is Conclusion.

GBDT and explicit solutions for the dynamical canonical system
Further we fix an initial Hamiltonian H ≥ 0, which (in the case H > 0) is equivalent to fixing the initial Hamiltonian H 0 = H −1 (see (1.5)). Given an initial m × m Hamiltonian H(x) ≥ 0 (x ≥ 0), each GBDT is (as usual) determined by some n × n matrices A and S(0) > 0 (n ∈ N) and by an n × m matrix Π(0) which satisfy the matrix identity (1.3), where j is defined in (1.1). Taking into account the initial values Π(0) and S(0) (and using the matrix A and Hamiltonian H(x)) we introduce matrix functions Π(x) and S(x) via the equations: It is easy to see [25] that (1.3) and (2.1) yield the identity Since S(0) > 0 and S ′ = ΠjHjΠ * ≥ 0, the matrices S(x) (x ≥ 0) are positive definite and invertible. The so called Darboux matrix from Darboux transformations is represented in GBDT (for each x ) as the transfer matrix function in Lev Sakhnovich [32][33][34] More precisely, for the case of the canonical system (i.e., for the case that (2.1) and (2.2) hold) we have [25]: respectively, where q 0 has the form (2.4). Then, the matrix functions P r o o f. It easily follows from (2.4) that q * 0 j + j q 0 = 0. Hence, formula (2.6) implies that u * ju ≡ j, and so According to (2.7) -(2.9) we have Differentiating Π * S −1 and taking into account (2.1) we obtain: Multiplying both sides of (2.2) by S −1 from the left and from the right, we derive Next, we substitute (2.12) into (2.11): (2.14) Finally, formula (1.1) is immediate from (2.7) and (2.14).
Remark 2.2 Comparing (2.3) and (2.6) we easily see that We note that w A (x, 0) and w A (0, 0) −1 are well-defined in the case det A = 0.
Let us consider in detail the case of the trivial initial Hamiltonian, that is, the case H(x) ≡ I m . In order to avoid confusion, we assume for this case that m 1 > 0 and m 2 > 0 in (1.1). (The modification of our considerations for m 1 > 0 and m 2 = 0 is simple and evident.) Partition Π(0) into two blocks Π(0) = ϑ 1 ϑ 2 (where ϑ i are n×m i matrices) and derive from (2.1) explicit expressions for Π(x) and S(x): (2.17) The following corollary of Theorem 2.1 presents explicit solutions of (1.1).
In the formula (2.17), the expression for Π is somewhat more explicit then the expression for S. However, the expression for S in (2.17) is easily rewritten in terms of the solutions C k of the matrix identities (linear algebraic equations): Indeed, in view of (2.19) and the first equality in (2.17) we have Using (2.20), we immediately simplify the expression for S in (2.17) and derive the corollary below. ) holds for k = 1, we may set C 2 = C 1 − S(0), and (2.19) will hold for k = 2 (with this choice of C 2 ) as well. If (2.19) holds for k = 2 we may set C 1 = C 2 + S(0). We note that the identities (2.19) always have unique solutions C k when σ(A) ∩ σ(A * ) = ∅.
Substituting the first equality in (2.17) and equality (2.21) (i.e., explicit expressions for Π(x) and S(x)) into (2.16), we obtain an explicit expression for u(x). Now, substituting explicit expressions for Π(x), S(x) and u(x) into (2.18), we obtain explicit expressions for H(x) and Y (x, t).
Explicit solutions of the related to the system (1.1) Loewner's system were constructed in [10]. However, the dependence of L on both variables is essential in that construction and the procedure itself is more complicated.
3 Analysis of the obtained solutions 1. In this section, we consider H and Y constructed in Theorem 2.1. According to [37, Section 2], the energy E Y h (t) of the solution Y (x, t)h (h ∈ C m ) on the interval 0 ≤ x ≤ a is given in physical models by the expression The vector h in (3.1) determines the boundary condition for the solution Y h: In view of (2.7) and (2.8) we have We note that the energy in (3.5) is directly expressed via the boundary values S(0) and S(a). Let us set Then, relation (2.7) implies that On the other hand, (2.12) yields the formula Formulas (3.5)-(3.8) imply the following equality, which is similar to the relations for supply rate for port-Hamiltonian systems that appear in the literature: Recall that S(x) > 0 and that S(x) is a monotonically nondecreasing matrix function. Hence, there is a finite nonnegative limit κ S of S(x) −1 when x tends to infiity.
Remark 3.2 When we consider the case of the semiaxis 0 ≤ x < ∞ instead of the interval 0 ≤ x ≤ a, we substitute κ S (instead of S(a) −1 ) into the expression (3.5) for energy.

2.
In this paragraph, we study the behavior of the explicit solution Y (x, t) given by (2.18). According to Corollary 2.3, Y is expressed in terms of Π(x) and S(x). Similar to [13,Sect. 6] (see also [28]) we include into consideration matrix functions Taking into account (2.2), (2.17) and (3.10) we have It is immediate from (3.10)-(3.12) that Q and R are positive-definite and monotonically nondecreasing matrix functions, and so the limits Q(x) −1 and R(x) −1 (x → ∞) exist. We set Relations (2.5), (3.10) and the first equality in (2.17) imply that w A (x, 0) admits representation (3.14) In view of (2.15), the asymptotics of w A (x, 0), which is given in the next proposition, provides the asymptotics of u in the expression (2.18) for Y .
(3.15) P r o o f. The equality (3.15) is immediate from (3.13), (3.14) and relations On the other hand, (3.11), (3.12) and the first equality in (2.17) yield Equality (3.21) and the first equality in (3.13) imply that the entries of Q −1 ′ are summable on [0, ∞) (i.e., Q −1 ′ is summable on [0, ∞)). Hence, taking into account that Q −1 is bounded, we see that the derivative of the left-hand side of (3.19) is summable, and so the derivative of the right-hand side of (3.19) is summable. Moreover, since Q −1 ′ is summable, formula (3.21) implies that the right-hand side of (3.19) is summable.
The fact that the right-hand side of (3.19) and its derivative are both summable on [0, ∞) means that the right-hand side of (3.19) tends to zero, that is, The first equality in (3.16) follows from (3.23). The second equality in (3.16) is proved in the same way using (3.20) and (3.22).
Taking into account (2.18) and the proposition above we show that the behavior of Y (x, t) is characterized by two exponents e i(t+x)A and e −i(t+x)A .

24)
where the functions "little-o" tend to 0 when x tends to infinity.
P r o o f. According to Proposition 3.3 and Remark 2.2, the first factor (in round brackets) on the right-hand side of (3.24) gives us u(x) * j.
Using the first equality in (2.17) and relations (3.10) and (3.13), we have (3.25) Hence, we rewrite Π(x) * S(x) −1 e itA in the form of the second factor on the right-hand side of (3.24). Now, (3.24) is immediate from (2.18). Namely, we obtain: Correspondingly, we rewrite (3.24) in the form, which is convinient for the study of asymptotics of Y with large x and t, when the spectrum of A belongs to the upper semiplane: Clearly, we can write an analog of (3.26) with R instead of Q as well.
3. We mentioned already in the Introduction that the linear similarity of jH(x) to some diagonal matrix function D(x) and the corresponding representation jH(x) = T (x)D(x)T (x) −1 is essential for well-posedness problems. In our case, using (2.9) we can rewrite (2.8) in the form Hence, if jH −1 admits a representation where D(x) is some diagonal matrix function, equalities (3.28) and (3.29) yield the following result.

Conclusion
The solutions of the dynamical canonical system, which are constructed in this note, and their behavior are of independent interest. These solutions may also serve as examples for some problems arising in control theory.
The results of the note are easily generalized for the case H = H * where the requirement H > 0 is omitted. We could also study (in the spirit of [28]) blow up solutions with singularities, which appear, if we omit the requirement S(0) > 0. We mention that GBDT was successfully applied for the construction of explicit solutions of nonlinear dynamical systems as well (see, e.g., various references in [10,26,29,32]).
In the control problems for (1.1) boundary conditions are essential. where W is an m×2m matrix, are considered in [16]. Choosing m-dimensional invariant subspaces L A of A and using (2.10), we can always construct such matrices W that (4.1) holds for all the solutions Y (x, t)h (h ∈ L A ), where Y is given by (2.7).
It is our plan to consider (in the next paper) Bäcklund-Darboux transformations for the case of a more general than (1.1) system ∂ ∂t Y (x, t) = P 1 ∂ ∂x + P 0 ) H(x)Y (x, t) , H(x) > 0, P 1 = P * 1 , (4.2) and study applications to control theory in a more detailed way.