Synchronizing chaotic PDE system using backstepping and its application to image encryption

This paper is concerned with the observer design problem for a nonlinear hyperbolic system with a modified van der Pol boundary condition in which an integral term is included. The observer plays a role of a chaotic synchronizing system when applying it to secure communication. In our previous work (Sano et al. Secure communication systems using distributed parameter chaotic synchronization. SICE Trans. 2021;57(2):78–85), a nonlinear hyperbolic system without the integral term was treated and a simple synchronizing system was constructed, where the constant coefficients of system played a role of encryption keys. But the keys were vulnerable from a safety standpoint. On the other hand, in the case where the integral term is included in the boundary condition, we need to construct observers. The weighted function contained in the integral term gives a new encryption key of distributed type. An application to image encryption is also discussed and numerical simulation results are given.


Introduction
We shall consider the nonlinear hyperbolic system with a modified van der Pol boundary condition ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ where γ (x) is a real-valued function such that γ ∈ C 1 [0, 1] and γ (0) = 0, and κ is a real constant. For a given z ∈ R, φ(z) expresses a solution y to the equation That is, φ(z) = y. For each z ∈ R, Equation (2) has a unique solution y ∈ R, if the conditions 0 < α ≤ 1 and β > 0 are satisfied (e.g. [1]). Hereafter, it is assumed that α and β are chosen such that 0 < α ≤ 1 and β > 0. If γ (x) ≡ 0, then we obtain the nonlinear hyperbolic system (1) from a wave equation with a van der Pol boundary condition, and u, v are the Riemann invariants of the wave equation; see, e.g. [1]. Using the method of characteristic lines, we can characterize the solution of system (1). First, we express the boundary condition of (1) as u(t, 1) = φ(v(t, 1)) + f (u(t, ·)), v(t, 0) = ψ(u(t, 0)), with γ (x) and κ being the same function and constant as in (1). Considering the reflection of wave at x = 0, 1, the solution (u, v) of (1) is determined as follows (see Appendix 1): For x ∈ [0, 1] and t = 2k + τ (k = 0, 1, 2, . . . , 0 ≤ τ < 2), where u k (x) and v k (x) are defined as with φ being implicitly defined by (2). In the above, we assume that the initial condition u 0 , v 0 ∈ H 1 (0, 1) satisfies the compatible conditions Although it is difficult to show the well-posedness of (1) within the framework of functional analysis, we can directly determine the solution.
For the wave equation with a van der Pol boundary condition not including an integral term, the observer design has been studied in [2]. In this paper, we first construct observers for (1) with γ (x) ≡ 0, and then apply them to image encryption as an example of secure communication. For lumped parameter systems, secure communication systems using chaotic synchronization have been investigated by many researchers since the mid-1990s. The common idea is the following: After communication information is embedded into a chaotic signal in the modulation component, it is sent to the receiving side, and, in the demodulation component, the communication information is restored by chaotic synchronization [3][4][5][6][7]. Here, chaotic synchronization is a phenomenon such that two systems whose states chaotically behave with the same dynamic characteristics are synchronized by adding a control input to one system. For example, it is accomplished by constructing observers. On the other hand, there are a few works concerning the design of secure communication systems using the chaos of PDEs [8][9][10]. In [9], it has been shown that two identical time-delayed Chua's circuits can be synchronized using a boundary control and that the synchronization can be applied to multichannel spread-spectrum communications. In [8], for two identical systems (1) with γ (x) ≡ 0, it has been shown that they can be synchronized by a simple control law, based on the method of characteristic line and that the synchronization is applicable to image encryption, where the constants α, β, κ of (1) and (2) play a role of encryption keys. In [10], an image encryption scheme has been proposed based on a hyperbolic system with nonlinear boundary conditions, which brings a chaotic phenomenon. In that paper, the solution of the hyperbolic system is used to shuffle the position of each pixel. However, the method of synchronize two identical hyperbolic systems is not considered. That is, in [10], the encryption and decryption are assumed to be done non-simultaneously.
In the case where an integral term is included in the boundary condition such as system (1), we need to construct observers for synchronization. Especially, we use backstepping design for first-order hyperbolic systems (see, e.g. [11][12][13][14][15][16][17][18][19]). The merit of using the integral term lies in the fact that the weighted function γ (x) gives an additional 'distributed' encryption key.

Observer design for synchronization
To estimate the states u(t, ·) and v(t, ·) of (1) from the output v(t, 1), we consider the 2 × 2 nonlinear hyperbolic system (3) is called a Luenberger-type observer. Introducing the error variablesũ := u −û andṽ := v −v, the following system is obtained: The main difference from [17,18] is that system (1) has the nonlinear function φ in the boundary condition at x = 1. However, a suitable choice of observers leads to a linear error system. Hence, one can apply the backstepping method to the error system (4) as in [17,18]. In particular, we use the Volterra-Fredholm integral transformation The problem is to determine the kernels p and q in (5) and the gains g and h in (4) so as to achieveũ(t, ·) → 0 andṽ(t, ·) → 0 as t goes to ∞. Differentiating (5) and performing integration by We here note the following facts: (i) If all the terms enclosed in { · } of the right-hand side of (6) are zero, Under the condition γ (0) = 0, we sequentially determine the kernels p and q and the gains g and h as follows: Step 1 -Design of p. 'Attention to (6) and (i), (ii)'. Let on D p . From (7), the solution is given by Step 2 -Design of q. 'Using p, find q '. 'Attention to (6) (8), the solution is given by Step 3 -Design of h. 'Using q, find h '. ' Attention to (4) and (iii)'. Determine the gain h(x), x ∈ [0, 1] such that the solutionṽ(t, ·) of the hyperbolic Equation (9) vanishes for all t ≥ 1.
Concretely, one can construct the gain h by solving Equations (10) and (11).
The derivation is shown in Appendix 2. Since it follows from (10) that r(x, y) = κq(0, y − x), the gain h is given by solving the integral Equation (11).
Step 4 -Design of g. 'Using p, q, h, find g '. 'Attention to (6) and (i)'. Find the solution g(x), x ∈ [0, 1] to the integral equation As a result, for the error system (4), the following system can be considered as a target system: For the upper part of system (13), we see that w(t, ·) becomes zero at t = 1, since Therefore, at t = 1, the lower part of system (13) equals the hyperbolic Equation (9). From the way of construction of the gain h in the Step 3,ṽ(t, ·) becomes zero at t = 2. On the other hand, it follows from (5) that From the invertibility [20] of the transformation T : we see thatũ(t, ·) andṽ(t, ·) vanish for t ≥ 2. Now, we introduce a Hilbert space X := L 2 (0, 1) × L 2 (0, 1) and its subspace Then, we have the following theorem:

Application to image encryption
In this section, we study the modulation/demodulation of image data with (M + 1) × (L + 1) pixel. We first need to discretize system (1) and synchronization system (3) For synchronization system (3), in the same fashion, . . .
To discretize these systems to spatial direction, we use the upwind difference scheme, that is, Further, for its time integration of u, v,û, andv, we use the Euler method with time mesh width t = x. Hereafter, we denote Equation (1) discretized system by (1) , and Equation (3) discretized system by (3) .

Remark 3.1:
The above discretization is the upwind difference scheme based on the characteristic line. Since t = x, the waves u andû travel along the characteristic line t + x = const , and the waves v andv travel along the characteristic line t − x = const.
In what follows, we state the concrete construction of system in the case where one transmits an image data from subsystem S 1 to subsystem S 2 . Modulation M : is assumed that, for arbitrarily fixed a, b, c ∈ R L+1 ,  G(a, b, c, ·) has the inverse G −1 (a, b, c, ·) and satisfies the following condition:  a, b, c, ξ 1 where | · | R L+1 denotes the usual Euclidean norm.
Especially, we choose G such that ω[k] of (14) behaves randomly and differently fromû[k] andv [k] so that the original image cannot be found at all in the modulated image {c 12 [k]}.
Demodulation  Figure 1 is the diagram when the synchronization system (3) is used. In the case where an image data of (M + 1) × (L + 1) pixel is transmitted from S 1 to S 2 , after a lapse of runup time, we have to set the original image data to s 1 [k] ∈ R L+1 row by row and perform the sending operation (M + 1) times. On the other hand, on the receiving side, the image data of (M + 1) × (L + 1) pixel is restored by stocking up with t 2 [k] ∈ R L+1 in sequence.
Next, we consider the case where the weighted function contained in the nonlocal boundary condition of system (1) fluctuates from γ (x) = 2 sin(7π x) to γ (x) = 1.95 sin(6.75π x) (see Figure 4). The function γ (x) contained in system (3) remains the same. Then, using the same observer gains g and h as in Figure 2, we had a numerical simulation result shown in Figure 5, where the same initial condition and the same runup time as in Figure 3 were used. We cannot find the original image at all from the restored image. In this case, the maximum of error between the original image {s 1 [k]} and the restored one {t 2 [k]} was 0.5689. From the right column of Figure 5, we see that the error system with statesũ andṽ is destabilizing due to a small discrepancy in the function γ (x).
In this paper, we have extended the G such that each argument of G can take a vector value.

Conclusion
In this paper, we proposed the observer design method for a nonlinear hyperbolic system with a modified van der Pol boundary condition including an integral term, and then, as an application example, we treated image encryption which was regarded as a kind of secure communication. The proposed method of image encryption includes three constant keys α, β, κ and one distributed key function γ (x), x ∈ [0, 1]. Indeed, since the γ (x) is approximated as γ (x i ), i = 0, 1, . . . , L, the proposed method will include very large number of constant keys compared to that of [8]. In this sense, the method proposed here can be said to be safer.
becomes the wave u, where, by the boundary condition at