In this paper we study backward stochastic differential equations with general terminal value and general random generator. In particular, we do not require the terminal value be given by a forward diffusion equation. The randomness of the generator does not need to be from a forward equation, either. Motivated from applications to numerical simulations, first we obtain the L p -Hölder continuity of the solution. Then we construct several numerical approximation schemes for backward stochastic differential equations and obtain the rate of convergence of the schemes based on the obtained L p -Hölder continuity results. The main tool is the Malliavin calculus.

In this paper we study a singular stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter H > 1 2 . Under some assumptions on the drift, we show that there is a unique solution, which has moments of all orders. We also apply the techniques of Malliavin calculus to prove that the solution has an absolutely continuous law at any time t > 0.

In this paper, an adaptive nonsingular terminal sliding mode control (ANTSMC) is investigated for attitude tracking of spacecraft with actuator faults. First, a nonsingular fast terminal sliding mode surface is designed to avoid the singularity. Finite-time attitude control is developed using the nonsingular terminal sliding mode technique, which can make the attitude and angular velocity tracking errors converge to zero in finite time in the presence of uncertainties and external disturbances. Second, the total uncertainty is deduced to be bounded. The adaptive laws are incorporated to develop the ANTSMC, removing the restriction on the upper bound of the lumped uncertainty. The finite-time convergence of the closed-loop system with ANTSMC is proved using the Lyapunov stability theory. Finally, the simulation results are presented to demonstrate the performance of the proposed controllers. INDEX TERMS Spacecraft, nonsingular terminal sliding mode, adaptive, finite time, attitude control, actuator faults.

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In this paper, we study spatial averages for the parabolic Anderson model in the Skorohod sense driven by rough Gaussian noise, which is colored in space and time. We include the case of a fractional noise with Hurst parameters H 0 in time and H 1 in space, satisfying H 0 ∈ (1/2, 1), H 1 ∈ (0, 1/2) and H 0 + H 1 > 3/4. Our main result is a functional central limit theorem for the spatial averages. As an important ingredient of our analysis, we present a Feynman-Kac formula that is new for these values of the Hurst parameters.

The data space for audio signals is large, the correlation is strong, and the traditional encryption algorithm cannot meet the needs of efficiency and safety. To solve this problem, an audio encryption algorithm based on Chen memristor chaotic system is proposed. The core idea of the algorithm is to encrypt the audio signal into the color image information. Most of the traditional audio encryption algorithms are transmitted in the form of noise, which makes it easy to attract the attention of attackers. In this paper, a special encryption method is used to obtain higher security. Firstly, the Fast Walsh–Hadamar Transform (FWHT) is used to compress and denoise the signal. Different from the Fast Fourier Transform (FFT) and the Discrete Cosine Transform (DCT), FWHT has good energy compression characteristics. In addition, compared with that of the triangular basis function of the Fast Fourier Transform, the rectangular basis function of the FWHT can be more effectively implemented in the digital circuit to transform the reconstructed dual-channel audio signal into the R and B layers of the digital image matrix, respectively. Furthermore, a new Chen memristor chaotic system solves the periodic window problems, such as the limited chaos range and nonuniform distribution. It can generate a mask block with high complexity and fill it into the G layer of the color image matrix to obtain a color audio image. In the next place, combining plaintext information with color audio images, interactive channel shuffling can not only weaken the correlation between adjacent samples, but also effectively resist selective plaintext attacks. Finally, the cryptographic block is used for overlapping diffusion encryption to fill the silence period of the speech signal, so as to obtain the ciphertext audio. Experimental results and comparative analysis show that the algorithm is suitable for different types of audio signals, and can resist many common cryptographic analysis attacks. Compared with that of similar audio encryption algorithms, the security index of the algorithm is better, and the efficiency of the algorithm is greatly improved.

In this paper, we consider fractional parabolic equation of the form ∂uα 2 with α ∈ (0, 2] is a fractional Laplacian andẆ is a Gaussian noise colored in space and time. The precise moment Lyapunov exponents for the Stratonovich solution and the Skorohod solution are obtained by using a variational inequality and a Feynman-Kac type large deviation result for space-time Hamiltonians driven by α-stable process. As a byproduct, we obtain the critical values for θ and η such that E exp θ

Electrohydraulic loading system is a torque servo system with high-accuracy and high-frequency response. In this paper, an adaptive extended state observer-based flatness nonlinear output control is proposed to improve the torque tracking performance of electrohydraulic loading system. This method combines a flatness concept, expected state, adaptive extended state observer and system output to develop a stable control system. Expected input feedforward based on the flatness property is designed to provide model compensation for bandwidth enhancement. An adaptive extended state observer is proposed to estimate the unmeasured states and the unmodeled dynamics. Based on estimated states and disturbances, state feedback control is developed to ensure the stability of closed-loop system, and to improve torque tracking accuracy and system robustness. The stability of the closed-loop system is proved by the Lyapunov stability theory. Extensive experiments were carried out to verify the performance of high-accuracy tracking of the proposed control strategy.