In this paper the exact internal controllability for a coupled system of wave equations with arbitrarily given coupling matrix is established. Based on this result, the exact internal synchronization and the exact internal synchronization by p-groups are successfully considered.

The authors are concerned with the sharp interface limit for an incompressible Navier-Stokes and Allen-Cahn coupled system in this paper. When the thickness of the diffuse interfacial zone, which is parameterized by ε, goes to zero, they prove that a solution of the incompressible Navier-Stokes and Allen-Cahn coupled system converges to a solution of a sharp interface model in the L ^∞(L ²) ∩ L ²(H ¹) sense on a uniform time interval independent of the small parameter ε. The proof consists of two parts: One is the construction of a suitable approximate solution and another is the estimate of the error functions in Sobolev spaces. Besides the careful energy estimates, a spectral estimate of the linearized operator for the incompressible Navier-Stokes and Allen-Cahn coupled system around the approximate solution is essentially used to derive the uniform estimates of the error functions. The convergence of the velocity is well expected due to the fact that the layer of the velocity across the diffuse interfacial zone is relatively weak.

Estimation and control problems with binary-valued observations exist widely in practical systems. However, most of the related works are devoted to finite impulse response (FIR for short) systems, and the theoretical problem of infinite impulse response (IIR for short) systems has been less explored. To study the estimation problems of IIR systems with binary-valued observations, the authors introduce a projected recursive estimation algorithm and analyse its global convergence properties, by using the stochastic Lyapunov function methods and the limit theory on double array martingales. It is shown that the estimation algorithm has similar convergence results as those for FIR systems under a weakest possible non-persistent excitation condition. Moreover, the upper bound for the accumulated regret of adaptive prediction is also established without resorting to any excitation condition.

From the mesoscopic point of view, a definition of soft point is introduced by considering the attributes of geometric profile and mass distribution. After that, this concept is used to develop the soft matching technique to simulate the chaotic behaviors of the equations. Especially, a tennis model with deformation factor a(t) is proposed to derive a generalized Newton-Stokes equation v′(t) = λ(v _T − a(t)v(t)). Furthermore, a concept of duality of deformation factor a(t) and velocity v(t) with respect to the generalized Newton-Stokes equation is established. To solve this equation, two data-driven models of a(t) are provided, one is based on the concept of soft matching, while the other is by using the amplitude modulation. Finally, the related iterative algorithm is developed to simulate the motion of the falling body via the duality of a(t) and v(t). Numerical examples successfully demonstrate the phenomenon of chaos, which consists of the continual random oscillations and sudden accelerations. Moreover, the algorithm is tested by using larger coefficients corresponding to the terminal velocity and shows more satisfactory results. It may enable us to characterize the total energy of the dynamical system more accurately.

The speeding-up and slowing-down (SUSD) direction is a novel direction, which is proved to converge to the gradient descent direction under some conditions. The authors propose the derivative-free optimization algorithm SUSD-TR, which combines the SUSD direction based on the covariance matrix of interpolation points and the solution of the trust-region subproblem of the interpolation model function at the current iteration step. They analyze the optimization dynamics and convergence of the algorithm SUSD-TR. Details of the trial step and structure step are given. Numerical results show their algorithm’s efficiency, and the comparison indicates that SUSD-TR greatly improves the method’s performance based on the method that only goes along the SUSD direction. Their algorithm is competitive with state-of-the-art mathematical derivative-free optimization algorithms.

A geometric intrinsic pre-processing algorithm(GPA for short) for solving large-scale discrete mathematical-physical PDE in 2-D and 3-D case has been presented by Sun (in 2022–2023). Different from traditional preconditioning, the authors apply the intrinsic geometric invariance, the Grid matrix G and the discrete PDE mass matrix B, stiff matrix A satisfies commutative operator BG = GB and AG = GA, where G satisfies G ^m = I, m ≪ dim(G). A large scale system solvers can be replaced to a more smaller block-solver as a pretreatment in real or complex domain.

In this paper, the authors expand their research to 2-D and 3-D mathematical physical equations over more wide polyhedron grids such as triangle, square, tetrahedron, cube, and so on. They give the general form of pre-processing matrix, theory and numerical test of GPA. The conclusion that “the parallelism of geometric mesh pre-transformation is mainly proportional to the number of faces of polyhedron” is obtained through research, and it is further found that “commutative of grid mesh matrix and mass matrix is an important basis for the feasibility and reliability of GPA algorithm”.

It is well-known that the general Manakov system is a 2-components nonlinear Schrödinger equation with 4 nonzero real parameters. The analytic property of the general Manakov system has been well-understood though it looks complicated. This paper devotes to exploring geometric properties of this system via the prescribed curvature representation in the category of Yang-Mills’ theory. Three models of moving curves evolving in the symmetric Lie algebras u(2,1) = k _α ⊕ m _α (α = 1, 2) and u(3) = k ₃ ⊕ m ₃ are shown to be simultaneously the geometric realization of the general Manakov system. This reflects a new phenomenon in geometric realization of a partial differential equation/system.

The authors prove error estimates for the semi-implicit numerical scheme of sphere-constrained high-index saddle dynamics, which serves as a powerful instrument in finding saddle points and constructing the solution landscapes of constrained systems on the high-dimensional sphere. Due to the semi-implicit treatment and the novel computational procedure, the orthonormality of numerical solutions at each time step could not be fully employed to simplify the derivations, and the computations of the state variable and directional vectors are coupled with the retraction, the vector transport and the orthonormalization procedure, which significantly complicates the analysis. They address these issues to prove error estimates for the proposed semi-implicit scheme and then carry out numerical experiments to substantiate the theoretical findings.

Modeling of frictional contacts is crucial for investigating mechanical perforances of composite materials under varying service environments. The paper considers a linear elasticity system with strongly heterogeneous coefficients and quasistatic Tresca friction law, and studies the homogenization theories under the frameworks of H-convergence and small ε-periodicity. The qualitative result is based on H-convergence, which shows the original oscillating solutions will converge weakly to the homogenized solution, while the author’s quantitative result provides an estimate of asymptotic errors in H ¹-norm for the periodic homogenization. This paper also designs several numerical experiments to validate the convergence rates in the quantitative analysis.