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This paper introduces a low-boom aircraft optimization design method guided by equivalent area distribution, which effectively improves the intuitiveness and refinement of inverse design. A gradient optimization method based on discrete adjoint equations is proposed to achieve the fast solution of the gradient information of target equivalent area distribution relative to design variables and to drive the aerodynamic shape update to the optimal solution. An optimization experiment is carried out based on a self-developed supersonic civil aircraft configuration with engines. The results show that the equivalent area distribution adjoint equation can accurately solve the gradient information. After optimization, the sonic boom level of the aircraft was reduced by 13.2 PLdB, and the drag coefficient was reduced by 60.75 counts. Moreover, the equivalent area distribution adjoint optimization method has outstanding advantages, such as high sensitivity and fast convergence speed, and can take both the low sonic boom and the low drag force of the aircraft into account, providing a powerful tool for the comprehensive optimization design of supersonic civil aircraft by considering sonic boom and aerodynamic force. [ABSTRACT FROM AUTHOR]

This paper derives necessary optimality conditions for a certain class of optimal control problems without linearity or convexity assumptions. The optimal control problem has a general objective function of integral type and a finite number of isoperimetric type constraints. For proving the main result derived in this paper, the Lagrange function and the control Hamiltonian are introduced and an adjoint differential equation is stated. In addition, we formulate some examples where the derived necessary optimality conditions are applied. [ABSTRACT FROM AUTHOR]

Pine wood nematode disease has become nowadays a major threat to forest ecosystems and caused huge economic and environmental losses to many countries and regions. Considering the influence of environmental factors such as temperature, drought, and humidity, as well as possible prevention strategies, including trunk injection, cutting infected pines, and insecticide application, in this paper, we develop a stochastic pine wilt disease model with prevention strategies, aiming to obtain the conditions for its near‐optimal control. The near‐optimality problem is constructed by decreasing the infected pine forests and beetle population while minimizing the costs of prevention strategies. We first verify that, under certain convexity conditions, the near‐minimum condition of the Hamiltonian function is sufficient for the near‐optimal control. Then, by utilizing the spike variational technique, we derive the necessary conditions of the near‐optimal control. Finally, we present an example to illustrate the validity of the main theoretical results, which indicates that reasonable prevention strategies could effectively reduce pine wilt infection, and therefore, to some extent, could protect the pine forests from pine wilt disease and reduce economic losses caused by this disease. [ABSTRACT FROM AUTHOR]

Featured Application: This work could be readily used in modeling contaminant transportation in homogeneous porous media with constant mean transport velocity, first-order decay and linear equilibrium sorption, e.g., the spread of a solute in water. In particular, the proposed method recovers the time-dependent pollution source strength from point measurements in a(n) (un)bounded domain. In this paper, we suggest a method for recovering the unknown time-dependent strength of a contaminant concentration source from measurements of the concentration inside an unbounded domain. This problem is formulated as a Cauchy parabolic inverse problem. For its efficient numerical processing, the problem is solved by reduction of the Cauchy problem to a Dirichet one on a bounded domain using the method of the fundamental (potential) solutions in combination with an adjoint equation technique. A numerical solution to this approach is explained. Next, by choosing the source strength in the form of a finite series of shape functions with unknown constant coefficients and using a linear-square method, the term concentration source is estimated. Computational simulations using model examples from water pollution are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the non-zero-sum differential game problem based on the delayed backward doubly stochastic differential equation. We discuss the case where the controlled system contains time-delayed variables, and the time-delayed variables are different. We construct a new kind of adjoint equation consisting of a doubly stochastic differential equation and three simple differential equations. Under appropriate premises for such equations, we deduce a stochastic maximum principle as a necessary condition and a verification theorem for the Nash equilibrium point by using the duality method and convex variation technique. As an application, we apply our result to a linear delayed backward doubly stochastic non-zero-sum game problem to verify the effectiveness of our results. [ABSTRACT FROM AUTHOR]

The connection between numerical methods for solving differential equations and machine learning has been revealed recently. Differential equations have been proposed as continuous analogues of deep neural networks, and then used in handling certain tasks, such as image recognition, where the training of a model includes learning the parameters of systems of ODEs from certain points along their trajectories. Treating this inverse problem of determining the parameters of a dynamical system that minimize the difference between data and trajectory by a gradient-based optimization method presents the solution of the adjoint equation as the continuous analogue of backpropagation that yields the appropriate gradients. The paper explores an abstract approach that can be used to construct a family of loss functions with the aim of fitting the solution of an initial value problem to a set of discrete or continuous measurements. It is shown, that an extension of the adjoint equation can be used to derive the gradient of the loss function as a continuous analogue of backpropagation in machine learning. Numerical evidence is presented that under reasonably controlled circumstances the gradients obtained this way can be used in a gradient descent to fit the solution of an initial value problem to a set of continuous noisy measurements, and a set of discrete noisy measurements that are recorded at uncertain times. [ABSTRACT FROM AUTHOR]

Large-scale inverse problems that require high-performance computing arise in various fields, including regional air quality studies. The paper focuses on parallel solutions of an emission source identification problem for a 2D advection–diffusion–reaction model where the sources are identified by heterogeneous measurement data. In the inverse modeling approach we use, a source identification problem is transformed to a quasi-linear operator equation with a sensitivity operator, which allows working in a unified way with heterogeneous measurement data and provides natural parallelization of numeric algorithms by concurrent calculation of the rows of a sensitivity operator matrix. The parallel version of the algorithm implemented with a message passing interface (MPI) has shown a 40× speedup on four Intel Xeon Gold 6248R nodes in an inverse modeling scenario for the Lake Baikal region. [ABSTRACT FROM AUTHOR]

This paper develops a novel limited-memory method to solve dynamic optimization problems. The memory requirements for such problems often present a major obstacle, particularly for problems with PDE constraints such as optimal flow control, full waveform inversion, and optical tomography. In these problems, PDE constraints uniquely determine the state of a physical system for a given control; the goal is to find the value of the control that minimizes an objective. While the control is often low dimensional, the state is typically more expensive to store. This paper suggests using randomized matrix approximation to compress the state as it is generated and shows how to use the compressed state to reliably solve the original dynamic optimization problem. Concretely, the compressed state is used to compute approximate gradients and to apply the Hessian to vectors. The approximation error in these quantities is controlled by the target rank of the sketch. This approximate first- and second-order information can readily be used in any optimization algorithm. As an example, we develop a sketched trust-region method that adaptively chooses the target rank using a posteriori error information and provably converges to a stationary point of the original problem. Numerical experiments with the sketched trust-region method show promising performance on challenging problems such as the optimal control of an advection-reaction-diffusion equation and the optimal control of fluid flow past a cylinder. [ABSTRACT FROM AUTHOR]

We study relaxed stochastic control problems where the state equation is a forward backward doubly stochastic differential equation with Poisson jumps, where the set of strict (classical) controls need not be convex and the diffusion coefficient and the generator coefficient depends on the terms control. In this paper, we introduce a new approach to solve this open problem, the main result is necessary conditions as well as a sufficient for optimality in the form of a relaxed maximum principle, with application to linear quadratic stochastic control problem with random jumps. [ABSTRACT FROM AUTHOR]

In this paper, considering the effect of environmental factors such as climate change, temperature and drought on the growth of forests and beetles as well as the pine forests death and economic loss caused by outbreaks of mountain pine beetles, we first develop a stochastic mountain pine beetle model with pesticide application, and then consider its near‐optimal control (NOC) problem. The problem NOC is formulated by decreasing the mountain pine beetles while keeping the cost of pesticide application to a minimum. By Pontryagin stochastic maximum principle, we establish some sufficient and necessary conditions of the near‐optimality. Some numerical simulations are carried out to illustrate the theoretical results, which indicate that the pesticide application could effectively control the number of beetles and relieve the pressure from the attacking of beetles on the pines. To some extent, this control strategy could protect the pine forests and reduce economic losses caused by the mountain pine beetles. [ABSTRACT FROM AUTHOR]

This paper focuses on a general model of a controlled stochastic differential equation with mixed delay in the state variable. Based on the Itô formula, stochastic analysis, convex analysis, and inequality technique, we obtain a semi-coupled forward-backward stochastic differential equation with mixed delay and mixed initial-terminal conditions and prove that such forward-backward system admits a unique adapted solution. The verification theorem for an optimal control of a system with mixed delay is established. The obtained results generalize and improve some recent results, and they are more easily verified and applied in practice. As an application, we conclude with finding explicitly the optimal consumption rate from the wealth process of a person given by a stochastic differential equation with mixed delay which fit into our general model. [ABSTRACT FROM AUTHOR]

Let (E , E) be an arbitrary measurable space. The paper first focuses on studying the partial derivative of a function f : P 2 , 0 (R d × E) → R defined on the space of probability measures μ over (R d × E , B (R d) ⊗ E) whose first marginal μ 1 ≔ μ (⋅ × E) has a finite second order moment. This partial derivative is taken with respect to q (d x , z) , where μ has the disintegration μ (d x d z) = q (d x , z) μ 2 (d z) with respect to its second marginal μ 2 (⋅) = μ (R d × ⋅). Simplifying the language, we will speak of the derivative with respect to the law μ conditioned to its second marginal. Our results extend those of the derivative of a function g : P 2 (R d) → R over the space of probability measures with finite second order moment by P.L. Lions (see Lions (2013)) but cover also as a particular case recent approaches considering E = R k and supposing the differentiability of f over P 2 (R d × R k) , in order to use the derivative ∂ μ f to define the partial derivative (∂ μ f) 1. The second part of the paper focuses on investigating a stochastic maximum principle, where the controlled state process is driven by a general mean-field stochastic differential equation with partial information. The control set is just supposed to be a measurable space, and the coefficients of the controlled system, i.e., those of the dynamics as well as of the cost functional, depend on the controlled state process X , the control v , a partial information on X , as well as on the joint law of (X , v). Through considering a new second-order variational equation and the corresponding second-order adjoint equation, and a totally new method to prove the estimate for the solution of the first-order variational equation, the optimal principle is proved through spike variation of an optimal control and with the help of the tailor-made form of second-order expansion. We emphasize that in our assumptions we do not need any regularity of the coefficients neither in the control variable nor with respect to the law of the control process. [ABSTRACT FROM AUTHOR]

This paper studies the risk-sensitive optimal control problem for a backward stochastic system. More precisely, we set up a necessary stochastic maximum principle for a risk-sensitive optimal control of this kind of equations. The control domain is assumed to be convex and the generator coefficient of such system is allowed to be depend on the control variable. As a preliminary step, we study the risk-neutral problem for which an optimal solution exists. This is an extension of initial control system to this type of problem, where the set of admissible controls is convex. An example to carried out to illustrate our main result of risk-sensitive control problem under linear stochastic dynamics with exponential quadratic cost function. [ABSTRACT FROM AUTHOR]

In this paper, we study a new type of differential game problems of backward stochastic differential delay equations under partial information. A class of time-advanced stochastic differential equations (ASDEs) is introduced as the adjoint process via duality relation. By means of ASDEs, we suggest the necessary and sufficient conditions called maximum principle for an equilibrium point of non-zero sum games. As an application, an economic problem is putted into our framework to illustrate the theoretical results. In terms of the maximum principle and some auxiliary filtering results, an equilibrium point is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we extend the Euler-Bernoulli beam theory for bending boundary value problem into mechanically coupled system. We follow the inverse approach to find the exerted force on two beams separated by elastic material. The theory was utilized in two ways: in the first approach, we calculate the force exerted on the beams using known values for the stiffness constant and measured values for the beam deflections. In the second method, we calculate the stiffness constant using a single known force and measured deflections. These problems are typically illposed problems whose solution does not depend continuously on the boundary data. To minimize the variational functional, we develop an iterative algorithm based on the system of three equations: the direct, adjoint, and control equations. Then, we present numerical examples to obtain the solutions. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the maximum principle for optimal controls of stochastic systems with jumps by introducing a new method of variation. The control is allowed to enter both diffusion and jump terms and the control domain need not be convex. [ABSTRACT FROM AUTHOR]

Summary: This paper addresses a modified singularity removal technique for the eigenvalue or optimal mode problems in pipe flow using a pseudospectral method. The current approach results in the linear stability operator to be devoid of any unstable physically spurious modes, and thus, it provides higher numerical stability during time‐based integration. The correctness of the numerical operator is established by calculating the known eigenvalues of pipe Poiseuille flow. Subsequently, the optimal modes are determined with Farrell's approach and compared with the existing literature. The usefulness of this approach is further demonstrated in the time‐based numerical integration of the linearized Navier‐Stokes operator for the adjoint method–based optimal mode determination. The numerical scheme is implemented with the radial velocity‐radial vorticity formulation. Even number of Chebyshev‐Lobatto grid points are distributed over the domain r∈[−1,1] omitting the centerline, which also efficiently provides higher resolution near the wall boundary. The boundary conditions are imposed with homogeneous wall boundary conditions, whereas the analytic nature of a proper set of base functions enforces correct centerline conditions. The resulting redundancy introduced in the process is eliminated with the proper usage of parity. [ABSTRACT FROM AUTHOR]

This paper considers the numerical solution of optimal control problems based on ODEs. We assume that an explicit Runge-Kutta method is applied to integrate the state equation in the context of a recursive discretization approach. To compute the gradient of the cost function, one may employ Automatic Differentiation (AD). This paper presents the integration schemes that are automatically generated when differentiating the discretization of the state equation using AD. We show that they can be seen as discretization methods for the sensitivity and adjoint differential equation of the underlying control problem. Furthermore, we prove that the convergence rate of the scheme automatically derived for the sensitivity equation coincides with the convergence rate of the integration scheme for the state equation. Under mild additional assumptions on the coefficients of the integration scheme for the state equation, we show a similar result for the scheme automatically derived for the adjoint equation. Numerical results illustrate the presented theoretical results. [ABSTRACT FROM AUTHOR]

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Near-optimization is as sensible and important as optimization for both theory and applications. This paper concerns dynamic near-optimization, or near-optimal controls, for systems governed by the Ito stochastic differential equations (SDEs), where both the drift and diffusion terms are allowed to depend on controls and the systems are allowed to be degenerate. Necessary and sufficient conditions for a control to be near-optimal are studied. It is shown that any near-optimal control nearly maximizes the ``H-function'' (which is a generalization of the usual Hamiltonian and is quadratic with respect to the diffusion coefficients) in some integral sense, and vice versa if certain additional concavity conditions are imposed. Error estimates for both the near-optimality of the controls and the near-maximum of the H-function are obtained, based on some delicate estimates of the adjoint processes. Examples are presented to demonstrate the results. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with the solutions to both forward and backward mean-field stochastic partial differential equation and the corresponding optimal control problem for mean-field stochastic partial differential equation. The authors first prove the continuous dependence theorems of forward and backward mean-field stochastic partial differential equations and show the existence and uniqueness of solutions to them. Then they establish necessary and sufficient optimality conditions of the control problem in the form of Pontryagin's maximum principles. To illustrate the theoretical results, the authors apply stochastic maximum principles to study the infinite-dimensional linear-quadratic control problem of mean-field type. Further, an application to a Cauchy problem for a controlled stochastic linear PDE of mean-field type is studied. [ABSTRACT FROM AUTHOR]

In this paper, we deal with an optimal control, where the system is driven by a mean-field forward-backward doubly stochastic differential equation with jumps diffusion. We assume that the set of admissible control is convex, and we establish a necessary as well as a sufficient optimality condition for such system. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of strict optimal controls for linear backward doubly stochastic differential equations (LBDSDEs in short), where the control domain and the cost function were assumed convex. The proof is based on strong convergence techniques for the associated linear backward doubly stochastic differential equations and Mazur's theorem. We derive also necessary and sufficient conditions for optimality for this control problem. This result is based on the convex optimization principle. [ABSTRACT FROM AUTHOR]

We consider in this paper mixed relaxed-singular stochastic control problems, where the control variable has two components, the first being measure-valued and the second singular. The control domain is not necessarily convex and the system is governed by a nonlinear stochastic differential equation, in which the measure-valued part of the control enters both the drift and the diffusion coefficients. We establish necessary optimality conditions of the Pontryagin maximum principle type satisfied by an optimal relaxed-singular control, which exist under general conditions on the coefficients. The proof is based on the strict singular stochastic maximum principle established by Bahlali-Mezerdi, Ekeland's variational principle and some stability properties of the trajectories and adjoint processes with respect to the control variable. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to present some of our recent results about the second-order necessary conditions for stochastic optimal controls with the control variable entering into both the drift and the diffusion terms. In particular, when the control region is convex, a pointwise second-order necessary condition for stochastic singular optimal controls in the classical sense is established, whereas when the control region is allowed to be nonconvex, we obtain a pointwise second-order necessary condition for stochastic singular optimal controls in the sense of the Pontryagin-type maximum principle. Unlike deterministic optimal control problems or stochastic optimal control problems with control-independent diffusions, there exist some essential difficulties in deriving the pointwise second-order necessary optimality conditions from the integral conditions when the controls act in the diffusion terms of the stochastic control systems. Some techniques from Malliavin calculus are employed to overcome these difficulties. Moreover, it is found that, in contrast to the first-order necessary conditions, the correction part of the solution to the second-order adjoint equation appears in the pointwise second-order necessary conditions whenever the diffusion term depends on the control variable, even if the control region is convex. [ABSTRACT FROM AUTHOR]

In the present paper, the Ulam stability of the first order linear dynamic equation and its adjoint equation on a time scale is established by using the integrating factor method. Meantime, two examples are provided to illustrate our main results. [ABSTRACT FROM AUTHOR]

This paper is the second part of our series of work to establish pointwise second- order necessary conditions for stochastic optimal controls. In this part, we consider the general cases, i.e., the control region is allowed to be nonconvex, and the control variable enters into both the drift and the diffusion terms of the control systems. By introducing four variational equations and four adjoint equations (which are quite different from the case of convex control constraint), we obtain the desired necessary conditions for stochastic singular optimal controls in the sense of the Pontryagin-type maximum principle. [ABSTRACT FROM AUTHOR]

This paper studies an optimal control problem for partially observed forward-backward stochastic control system with a convex control domain and the forward diffusion term containing control variable. A maximum principle is proved for this kind of partially observable optimal control problems and the corresponding adjoint processes are characterized by the solutions of certain forward-backward stochastic differential equations in finite-dimensional spaces. One partially observed recursive linear-quadratic (LQ) optimal control example is also given to show the application of the obtained maximum principle. An explicit observable optimal control is obtained in this example. [ABSTRACT FROM AUTHOR]

The paper is concerned with a stochastic optimal control problem where the controlled systems are driven by Teugel’s martingales and an independent multi-dimensional Brownian motion. Necessary and sufficient conditions for an optimal control of the control problem with the control domain being convex are proved by the classical method of convex variation, and the coefficients appearing in the systems are allowed to depend on the control variables. As an application, the linear quadratic stochastic optimal control problem is studied. [ABSTRACT FROM AUTHOR]

The optimum planning of industrial pollutant sources, which optimizes the economic object without violating environmental constraints, is an important and hard task to be conquered. In this paper, an adjoint method is developed to solve the problem. The penalty function is introduced to deal with the environmental inequality constraints, and Lagrange function is constructed to derive the adjoint equation and the gradient of the object function. In this means, the gradient of the object function can be calculated by solving the adjoint equation, and the information from the gradient is used to make the object function descend and approach to an optimal solution after some iterations. A two-dimensional, simplified model is used for numerical experiments. The theoretical derivations are verified by the results of the experiments. Furthermore, the adjoint method is shown to be of excellent convergence and efficiency, which is adaptive to the fast development of air quality numerical models and super computers. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical solution of optimal control problems for ODEs. The methods considered here rely on some standard optimization code to solve a discretized version of the control problem under consideration. We aim to make available to the optimization software not only the discrete objective functional, but also its gradient. The objective gradient can be computed either from forward (sensitivity) information or backward (adjoint) information. The purpose of this paper is to discuss various ways of adjoint computation. It will be shown both theoretically and numerically that methods based on the continuous adjoint equation require a careful choice of both the integrator and gradient assembly formulas in order to obtain a gradient consistent with the discretized control problem. Particular attention is given to automatic differentiation techniques which generate automatically a suitable integrator. [ABSTRACT FROM AUTHOR]