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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]

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]

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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]

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 is devoted to the analysis of adjoint pairs of regular differential-algebraic equations with arbitrarily high tractability index. We consider both standard form DAEs and DAEs with properly involved derivative. We introduce the notion of factorization-adjoint pairs and show their common structure including index and characteristic values. We precisely describe the relations between the so-called inherent explicit regular ODE (IERODE) and the essential underlying ODEs (EUODEs) of a regular DAE. We prove that among the EUODEs of an adjoint pair of regular DAEs there are always those which are adjoint to each other. Moreover, we extend the Lyapunov exponent theory to DAEs with arbitrarily high index and establish the general class of DAEs being regular in Lyapunov's sense. The Perron identity which is well known in the ODE theory does not hold in general for adjoint pairs of Lyapunov regular DAEs. We establish criteria for the Perron identity to be valid. Examples are also given for illustrating the new results. [ABSTRACT FROM AUTHOR]

In this paper We consider a problem of optimal design in 2D for the wave equation with Dirichlet boundary conditions. We introduce a finite element discrete version of this problem in which the domains under consideration are polygons defined on the numerical mesh. We prove that, as the mesh size tends to zero, any limit, in the sense of the complementary-Hausdorff convergence, of discrete optimal shapes is an optimal domain for the continuous optimal design problem. We work in the functional and geometric setting introduced by V. Šveràk in which the domains under consideration are assumed to have an a priori limited number of holes. We present in detail a numerical algorithm and show the efficiency of the method through various numerical experiments. [ABSTRACT FROM AUTHOR]

This paper develops a variant of the Least Squares Shadowing (LSS) method, which has successfully computed the derivative for several chaotic ODEs and PDEs. The development in this paper aims to simplify Least Squares Shadowing method by improving how time dilation is treated. Instead of adding an explicit time dilation term as in the original method, the new variant uses windowing, which can be more efficient and simpler to implement, especially for PDEs. [ABSTRACT FROM AUTHOR]

In this paper, we consider optimal control problems governed by linear unsteady fluid-structure interaction problems. Based on a novel symmetric monolithic formulation, we derive optimality systems and provide regularity results for optimal solutions. The proposed formulation allows for natural application of gradient-based optimization algorithms and for space-time finite element discretizations. [ABSTRACT FROM AUTHOR]

In this paper, we deal with near optimality for stochastic control problems where the controlled system is described by a linear fully coupled forward–backward stochastic differential equation.We assume that the forward diffusion coefficient depends explicitly on the control variable and the control domain is not necessarily convex.We establish necessary as well as sufficient conditions for near optimality, satisfied by all near optimal controls. The proof of the main result is based on Ekeland’s variational principle and some estimates on the state and the adjoint processes with respect to the control variable. Finally an example which illustrate our results is presented. [ABSTRACT FROM AUTHOR]

The heavy sediment load of the Yellow River makes it difficult to simulate sediment concentration using classic numerical models. In this paper, on the basis of the classic one-dimensional numerical model of open channel flow, a variational-based data assimilation method is introduced to improve the simulation accuracy of sediment concentration and to estimate parameters in sediment carrying capacity. In this method, a cost function is introduced first to determine the difference between the sediment concentration distributions and available field observations. A one-dimensional suspended sediment transport equation, assumed as a constraint, is integrated into the cost function. An adjoint equation of the data assimilation system is used to solve the minimum problem of the cost function. Field data observed from the Yellow River in 2013 are used to test the proposed method. When running the numerical model with the data assimilation method, errors between the calculations and the observations are analyzed. Results show that (1) the data assimilation system can improve the prediction accuracy of suspended sediment concentration; (2) the variational inverse data assimilation is an effective way to estimate the model parameters, which are poorly known in previous research; and (3) although the available observations are limited to two cross sections located in the central portion of the study reach, the variational-based data assimilation system has a positive effect on the simulated results in the portion of the model domain in which no observations are available. [ABSTRACT FROM AUTHOR]

This paper is the first part of our series of work to establish pointwise second-order necessary conditions for stochastic optimal controls. In this part, both drift and diffusion terms may contain the control variable but the control region is assumed to be convex. Under some assumptions in terms of the Malliavin calculus, we establish the desired necessary conditions for stochastic singular optimal controls in the classical sense. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish the first and second order necessary conditions for stochastic optimal controls in infinite dimensions. The control system is governed by a stochastic evolution equation, in which both drift and diffusion terms may contain the control variable and the set of controls is allowed to be nonconvex. Only one adjoint equation is introduced to derive the first order necessary optimality condition either by means of the classical variational analysis approach or, under an additional assumption, by using differential calculus of set-valued maps. More importantly, in order to avoid the essential difficulty with the well-posedness of higher order adjoint equations, using again the classical variational analysis approach, only the first and the second order adjoint equations are needed to formulate the second order necessary optimality condition, in which the solutions to the second order adjoint equation are understood in the sense of the relaxed transposition. [ABSTRACT FROM AUTHOR]

• A stochastic chemostat model with non-zero cost inhibiting is proposed and investigated. • Some error estimations of the near-optimality about adjoint equation are obtained. • Sufficient and necessary conditions for stochastic near-optimal controls are established. • Numerical simulations and some conclusions are given. Near-optimal controls are as important as optimal controls for both theory and applications. Meanwhile, using inhibitor to control harmful microorganisms and ensure maximum growth of beneficial microorganisms (target microorganisms) is a very interesting topic in the chemostat. Thus, in this paper, we consider a stochastic chemostat model with non-zero cost inhibiting in finite time. The near-optimal control problem was constructed by minimizing the number of harmful microorganisms and minimizing the cost of inhibitor. We find that the Hamiltonian function is key to estimate objective function, and according to the adjoint equation, we obtain some error estimations of the near-optimality. Finally, we establish sufficient and necessary conditions for stochastic near-optimal controls of this model and numerical simulations and some conclusions are given. [ABSTRACT FROM AUTHOR]

This paper is devoted to the investigation of the second-order necessary conditions of singular control problems for controlled forward-backward delay systems in the case where the control domain is non-convex and the diffusion coefficients are independent of control. A second-order matrix-valued adjoint system is constructed to solve the obstacle caused by the product of the solution and its delay counterpart of the first-order variational equations. With the help of this observation, the second-order expansion of cost functional Y ε and the stochastic maximum principle are proved. Our work generalizes the classical case to the delay case [ABSTRACT FROM AUTHOR]

In this paper, we discuss the problem of discrete-time stochastic control with multiplicative noise, where the control domains may not be convex. We propose a novel approach inspired by the classical spike variation in continuous-time cases, but adapted to the discrete-time setting by perturbing in a random scale instead of time scale. Our approach allows us to obtain the maximum principle recursively without the need for variation equations or adjoint equations, which are typically required in previous methods. Moreover, we provide two maximum principles, one for the general case and the other when feedback forms exist. Our approach covers the previous results when the control domains are convex. Finally, we demonstrate the effectiveness of our new results through an example. [ABSTRACT FROM AUTHOR]

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.

Grid adaptation is an important strategy to improve the accuracy of output functions (e.g. drag, lift, etc.) in computational fluid dynamics (CFD) analysis and design applications. This paper presents a modified robust grid adaptation and error correction method for reducing simulation errors in integral outputs. The procedure is based on discrete adjoint optimization theory in which the estimated global error of output functions can be directly related to the local residual error. According to this relationship, local residual error contribution can be used as an indicator in a grid adaptation strategy designed to generate refined grids for accurately estimating the output functions. This grid adaptation and error correction method is applied to subsonic and supersonic simulations around three-dimensional configurations. Numerical results demonstrate that the sensitive grids to output functions are detected and refined after grid adaptation, and the accuracy of output functions is obviously improved after error correction. The proposed grid adaptation and error correction method is shown to compare very favorably in terms of output accuracy and computational efficiency relative to the traditional featured-based grid adaptation. [ABSTRACT FROM AUTHOR]

We present an analysis and numerical study of an optimal control problem for the Landau--de Gennes (LdG) model of nematic liquid crystals (LCs), which is a crucial component in modern technology. They exhibit long range orientational order in their nematic phase, which is represented by a tensor-valued (spatial) order parameter Q=Q(x). Equilibrium LC states correspond to Q functions that (locally) minimize an LdG energy functional. Thus, we consider an L2-gradient flow of the LdG energy that allows for finding local minimizers and leads to a semilinear parabolic PDE, for which we develop an optimal control framework. We then derive several a priori estimates for the forward problem, including continuity in space-time, that allow us to prove existence of optimal boundary and external "force" controls and to derive optimality conditions through the use of an adjoint equation. Next, we present a simple finite element scheme for the LdG model and a straightforward optimization algorithm. We illustrate optimization of LC states through numerical experiments in two and three dimensions that seek to place LC defects (where Q(x) = 0) in desired locations, which is desirable in applications. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish the first and second order necessary optimality conditions for stochastic optimal controls using the classical variational analysis approach. The control system is governed by a stochastic differential equation, in which both drift and diffusion terms may contain the control variable and the set of controls is allowed to be nonconvex. Only one adjoint equation is introduced to derive the first order necessary condition; while only two adjoint equations are needed to state the second order necessary conditions for stochastic optimal controls. [ABSTRACT FROM AUTHOR]

Bayesian inference coupled with computational fluid dynamics (CFD) has been widely used in the source term estimation (STE). At present, most scholars have studied the pollution source released in the time-invariant flow field and the turn-on time of the source is often regarded as a known parameter which requires no estimation. Nevertheless, it is of greater practical significance to estimate the source parameters by considering the time-varying characteristics of flow field. Besides, estimating the turn-on time is very important as it provides the critical information for post-disaster rescue and accident cause analysis. In this paper, the predicted concentrations are calculated by the adjoint equation in the time-varying flow field, and the turn-on time, taken as an unknown parameter, is estimated together with the source location and the release rate. Accurate estimated results are thereby obtained. Then, how the sensors' sampling time intervals and the measured data in different dispersion stages influence the estimated results is investigated. It is found that reducing the sampling time interval can decrease the uncertainties of the estimations. In addition, in order to estimate the turn-on time accurately, the measured information in the developing stage of dispersion is indispensable. However, only using the measured information in the stable stage of dispersion cannot predict the turn-on time. Finally, the earliest time when the source parameters can be estimated with accuracy is also explored. The results show that the method of the STE proposed can obtain accurate source information at the early stage of dispersion. • The source parameters are estimated in the time-varying flow field. • The turn-on time treated as an unknown parameter is estimated. • The predicted concentrations are obtained by adjoint equation in time-varying flow. • The impacts of key factors of measured data on the estimated results are studied. • The source parameters can be estimated accurately at early stage of dispersion. [ABSTRACT FROM AUTHOR]

This paper presents an adjoint method for locating potential leakage in a single-phase fluid pipeline based on the analytic solution of inversion modeling. By studying the mechanism of pipeline leakage pressure, the adjoint equation based on the governing equation of transient flow is established in the single-liquid phase aspect using inverse adjoint theory and sensitivity analysis method. The inverse transient adjoint equation is primarily derived from the single linear fluid pipeline in the semi-infinite domain. The Laplace method is then used to obtain an analytical solution that determines the location of pipeline leakage. The experimental results indicate that the analytic solution can quickly and accurately judge the leakage location of the pipeline. Furthermore, it presents a new approach to engineering applications, such as gas-liquid two-phase flow complex pipe networks, etc.