In the present paper, we show the some properties of the fuzzy R-solution of the control linear fuzzy differential inclusions and research the time-optimal problems for it.
For unachievable tracking problems, where the system output cannot precisely track a given reference, achieving the best possible approximation for the reference trajectory becomes the objective. This study aims to in...For unachievable tracking problems, where the system output cannot precisely track a given reference, achieving the best possible approximation for the reference trajectory becomes the objective. This study aims to investigate solutions using the Ptype learning control scheme. Initially, we demonstrate the necessity of gradient information for achieving the best approximation.Subsequently, we propose an input-output-driven learning gain design to handle the imprecise gradients of a class of uncertain systems. However, it is discovered that the desired performance may not be attainable when faced with incomplete information.To address this issue, an extended iterative learning control scheme is introduced. In this scheme, the tracking errors are modified through output data sampling, which incorporates lowmemory footprints and offers flexibility in learning gain design.The input sequence is shown to converge towards the desired input, resulting in an output that is closest to the given reference in the least square sense. Numerical simulations are provided to validate the theoretical findings.展开更多
A bicubic B-spline finite element method is proposed to solve optimal control problems governed by fourth-order semilinear parabolic partial differential equations.Its key feature is the selection of bicubic B-splines...A bicubic B-spline finite element method is proposed to solve optimal control problems governed by fourth-order semilinear parabolic partial differential equations.Its key feature is the selection of bicubic B-splines as trial functions to approximate the state and costate variables in two space dimensions.A Crank-Nicolson difference scheme is constructed for time discretization.The resulting numerical solutions belong to C2in space,and the order of the coefficient matrix is low.Moreover,the Bogner-Fox-Schmit element is considered for comparison.Two numerical experiments demonstrate the feasibility and effectiveness of the proposed method.展开更多
In this paper, the optimal control problem of parabolic integro-differential equations is solved by gradient recovery based two-grid finite element method. Piecewise linear functions are used to approximate state and ...In this paper, the optimal control problem of parabolic integro-differential equations is solved by gradient recovery based two-grid finite element method. Piecewise linear functions are used to approximate state and co-state variables, and piecewise constant function is used to approximate control variables. Generally, the optimal conditions for the problem are solved iteratively until the control variable reaches error tolerance. In order to calculate all the variables individually and parallelly, we introduce a gradient recovery based two-grid method. First, we solve the small scaled optimal control problem on coarse grids. Next, we use the gradient recovery technique to recover the gradients of state and co-state variables. Finally, using the recovered variables, we solve the large scaled optimal control problem for all variables independently. Moreover, we estimate priori error for the proposed scheme, and use an example to validate the theoretical results.展开更多
In this paper, time-optimal control problem for a liner n× n co-operative parabolic system involving Laplace operator is considered. This problem is, steering an initial state y(0)=u?, with control u?so that an o...In this paper, time-optimal control problem for a liner n× n co-operative parabolic system involving Laplace operator is considered. This problem is, steering an initial state y(0)=u?, with control u?so that an observation y(t) hitting a given target set in minimum time. First, the existence and uniqueness of solutions of such system under conditions on the coefficients are proved. Afterwards necessary and sufficient conditions of optimality are obtained. Finally a scaler case is given.展开更多
If A： D（A） X→X is a densely defined and closed linear operator, which generates a linear semigroup S （t） in Banach space X. The nonlocal control/ability for the following nonlocal semilinear problems： u＇ （t...If A： D（A） X→X is a densely defined and closed linear operator, which generates a linear semigroup S （t） in Banach space X. The nonlocal control/ability for the following nonlocal semilinear problems： u＇ （t） = Au （t） ＋ Bx（ t） ＋ f（ t, u（t） ）, 0≤t ≤ T with nonlocal initial condition u（0） = u0 ＋ g（u） is discussed in Banach space X. The results show that if semigroup S（t） is strongly continuous, the functionsf and g are compact and the control B is bounded, then it is nonlocally controllable. The nonlocal controllability for the above nonlocal problem is also studied when B and W are unbounded and the semigroup S（t） is compact or strongly continuous. For illustration, a partial differential equation is worked out.展开更多
Motivated by recent advances made in the study of dividend control and risk management problems involving the U.S.bankruptcy code,in this paper we follow[44]to revisit the De Finetti dividend control problem under the...Motivated by recent advances made in the study of dividend control and risk management problems involving the U.S.bankruptcy code,in this paper we follow[44]to revisit the De Finetti dividend control problem under the reorganization process and the regulator's intervention documented in U.S.Chapter 11 bankruptcy.We do this by further accommodating the fixed transaction costs on dividends to imitate the real-world procedure of dividend payments.Incorporating the fixed transaction costs transforms the targeting optimal dividend problem into an impulse control problem rather than a singular control problem,and hence computations and proofs that are distinct from[44]are needed.To account for the financial stress that is due to the more subtle concept of Chapter 11 bankruptcy,the surplus process after dividends is driven by a piece-wise spectrally negative Lévy process with endogenous regime switching.Some explicit expressions of the expected net present values under a double barrier dividend strategy,new to the literature,are established in terms of scale functions.With the help of these expressions,we are able to characterize the optimal strategy among the set of admissible double barrier dividend strategies.When the tail of the Lévy measure is log-convex,this optimal double barrier dividend strategy is then verified as the optimal dividend strategy,solving our optimal impulse control problem.展开更多
In the optimal control problem of nonlinear dynamical system,the Hamiltonian formulation is useful and powerful to solve an optimal control force.However,the resulting Euler-Lagrange equations are not easy to solve,wh...In the optimal control problem of nonlinear dynamical system,the Hamiltonian formulation is useful and powerful to solve an optimal control force.However,the resulting Euler-Lagrange equations are not easy to solve,when the performance index is complicated,because one may encounter a two-point boundary value problem of nonlinear differential algebraic equations.To be a numerical method,it is hard to exactly preserve all the specified conditions,which might deteriorate the accuracy of numerical solution.With this in mind,we develop a novel algorithm to find the solution of the optimal control problem of nonlinear Duffing oscillator,which can exactly satisfy all the required conditions for the minimality of the performance index.A new idea of shape functions method(SFM)is introduced,from which we can transform the optimal control problems to the initial value problems for the new variables,whose initial values are given arbitrarily,and meanwhile the terminal values are determined iteratively.Numerical examples confirm the high-performance of the iterative algorithms based on the SFM,which are convergence fast,and also provide very accurate solutions.The new algorithm is robust,even large noise is imposed on the input data.展开更多
This article presents the Parametric Iteration Method (PIM) for finding optimal control and its corresponding trajectory of linear systems. Without any discretization or transformation, PIM provides a sequence of func...This article presents the Parametric Iteration Method (PIM) for finding optimal control and its corresponding trajectory of linear systems. Without any discretization or transformation, PIM provides a sequence of functions which converges to the exact solution of problem. Our emphasis will be on an auxiliary parameter which directly affects on the rate of convergence. Comparison of PIM and the Variational Iteration Method (VIM) is given to show the preference of PIM over VIM. Numerical results are given for several test examples to demonstrate the applicability and efficiency of the method.展开更多
Software Defined Network(SDN)has been developed rapidly in technology and popularized in application due to its efficiency and flexibility in network management.In multi-controller SDN architecture,the Controller Plac...Software Defined Network(SDN)has been developed rapidly in technology and popularized in application due to its efficiency and flexibility in network management.In multi-controller SDN architecture,the Controller Placement Problem(CPP)must be solved carefully as it directly affects the whole network performance.This paper proposes a Multi-objective Greedy Optimized K-means Algorithm(MGOKA)to solve this problem to optimize worst-case and average delay between switches and controllers as well as synchronization delay and load balance among controllers for Wide Area Networks(WAN).MGOKA combines the process of network partition based on the K-means algorithm with cluster fusion based on the greedy algorithm and designs a normalization strategy to convert a multi-objective into a single-objective optimization problem.The simulation results depict that in different network scales with different numbers of controllers,the relative optimization rate of our proposed algorithm compared with K-means,K-means++,and GOKA can reach up to 101.5%,109.9%,and 79.8%,respectively.Moreover,the error rate between MGOKA and the global optimal solution is always less than 4%.展开更多
In this paper, we consider a fully discrete finite element approximation for time fractional optimal control problems. The state and adjoint state are approximated by triangular linear fi nite elements in space and &l...In this paper, we consider a fully discrete finite element approximation for time fractional optimal control problems. The state and adjoint state are approximated by triangular linear fi nite elements in space and <em>L</em>1 scheme in time. The control is obtained by the variational discretization technique. The main purpose of this work is to derive the convergence and superconvergence. A numerical example is presented to validate our theoretical results.展开更多
Dear Editor,In this letter,a constrained networked predictive control strategy is proposed for the optimal control problem of complex nonlinear highorder fully actuated(HOFA)systems with noises.The method can effectiv...Dear Editor,In this letter,a constrained networked predictive control strategy is proposed for the optimal control problem of complex nonlinear highorder fully actuated(HOFA)systems with noises.The method can effectively deal with nonlinearities,constraints,and noises in the system,optimize the performance metric,and present an upper bound on the stable output of the system.展开更多
In this paper, we propose the nonconforming virtual element method (NCVEM) discretization for the pointwise control constraint optimal control problem governed by elliptic equations. Based on the NCVEM approximation o...In this paper, we propose the nonconforming virtual element method (NCVEM) discretization for the pointwise control constraint optimal control problem governed by elliptic equations. Based on the NCVEM approximation of state equation and the variational discretization of control variables, we construct a virtual element discrete scheme. For the state, adjoint state and control variable, we obtain the corresponding prior estimate in H<sup>1</sup> and L<sup>2</sup> norms. Finally, some numerical experiments are carried out to support the theoretical results.展开更多
Advanced Process Control （APC） is necessary for oil refining and chemical process in China, but some problems have emerged in the application of APC techniques in this field. This paper discusses the conditions of A...Advanced Process Control （APC） is necessary for oil refining and chemical process in China, but some problems have emerged in the application of APC techniques in this field. This paper discusses the conditions of APC application concerning process design, distributed control system （DCS） choice and regular control. It analyzes the problems and strategies in APC application. Some suggestions are proposed for the enterprise to benefit from APC application.展开更多
We present in this paper a survey of recent results on the relation between time and norm optimality for linear systems and the infinite dimensional version of Pontryagin＇s maximum principle. In particular, we discus...We present in this paper a survey of recent results on the relation between time and norm optimality for linear systems and the infinite dimensional version of Pontryagin＇s maximum principle. In particular, we discuss optimality （or nonoptimality） of singular controls satisfying the maximum principle and smoothness of the costate in function of smoothness of the target.展开更多
Fast and high fidelity quantum control is the key technology of quantum computing. The hybrid system composed of the nitrogen-vacancy center and nearby Carbon-13 nuclear spin is expected to solve this problem. The nit...Fast and high fidelity quantum control is the key technology of quantum computing. The hybrid system composed of the nitrogen-vacancy center and nearby Carbon-13 nuclear spin is expected to solve this problem. The nitrogen-vacancy center electron spin enables fast operations for its strong coupling to the control field, whereas the nuclear spins preserve the coherence for their weak coupling to the environment. In this paper, we describe a strategy to achieve time-optimal control of the Carbon-13 nuclear spin qubit by alternating controlling the nitrogen-vacancy center electron spin as an actuator. We transform the qubit gate operation into a switched system. By using the maximum principle, we study the minimum time control of the switched system and obtain the time-optimal control of the qubit gate operation. We show that the X gate and Y gate operations are within 10μs while the fidelity reaches 0.995.展开更多
Iterative methods for solving discrete optimal control problems are constructed and investigated. These discrete problems arise when approximating by finite difference method or by finite element method the optimal co...Iterative methods for solving discrete optimal control problems are constructed and investigated. These discrete problems arise when approximating by finite difference method or by finite element method the optimal control problems which contain a linear elliptic boundary value problem as a state equation, control in the righthand side of the equation or in the boundary conditions, and point-wise constraints for both state and control functions. The convergence of the constructed iterative methods is proved, the implementation problems are discussed, and the numerical comparison of the methods is executed.展开更多
One of the key features required to realize fault-tolerant quantum computation is the robustness of quantum gates against errors.Since geometric quantum gate is naturally insensitivity to noise,it appears to be a prom...One of the key features required to realize fault-tolerant quantum computation is the robustness of quantum gates against errors.Since geometric quantum gate is naturally insensitivity to noise,it appears to be a promising routine to achieve high-fidelity,robust quantum gates.The implementation of geometric quantum gate however faces some troubles such as its complex interaction among multiple energy levels.Moreover,traditional geometric schemes usually take more time than equivalent dynamical ones.Here,we experimentally demonstrate a geometric gate scheme with the time-optimal control(TOC)technique in a superconducting quantum circuit.With a transmon qubit and operations restricted to two computational levels,we implement a set of geometric gates which exhibit better robustness features against control errors than the dynamical counterparts.The measured fidelities of TOC X gate and X/2 gate are 99.81%and 99.79%respectively.Our work shows a promising routine toward scalable fault-tolerant quantum computation.展开更多
A kind of direct methods is presented for the solution of optimal control problems with state constraints. These methods are sequential quadratic programming methods. At every iteration a quadratic programming which i...A kind of direct methods is presented for the solution of optimal control problems with state constraints. These methods are sequential quadratic programming methods. At every iteration a quadratic programming which is obtained by quadratic approximation to Lagrangian function and linear approximations to constraints is solved to get a search direction for a merit function. The merit function is formulated by augmenting the Lagrangian function with a penalty term. A line search is carried out along the search direction to determine a step length such that the merit function is decreased. The methods presented in this paper include continuous sequential quadratic programming methods and discreate sequential quadratic programming methods.展开更多
In this paper, the naturally evolving complex systems, such as biotic and social ones, are considered. Focusing on their structures, a feature is noteworthy, i.e., the similarity in structures. The relations between t...In this paper, the naturally evolving complex systems, such as biotic and social ones, are considered. Focusing on their structures, a feature is noteworthy, i.e., the similarity in structures. The relations between the functions and behaviors of these systems and their similar structures will be studied. Owing to the management of social systems and the course of evolution of biotic systems may be regarded as control processes, the researches will be within the scope of control problems. Moreover, since it is difficult to model for biotic and social systems, it will start with the control problems of complex systems, possessing similar structures, in engineering. The obtained results show that for either linear or nonlinear systems and for a lot of control problems similar structures lead to a series of simplifications. In general, the original system may be decomposed into reduced amount of subsystems with lower dimensions and simpler structures. By virtue of such subsystems, the control problems of original system can be solved more simply. At last, it turns round to observe the biotic and social systems and some analyses are given.展开更多
文摘In the present paper, we show the some properties of the fuzzy R-solution of the control linear fuzzy differential inclusions and research the time-optimal problems for it.
基金supported by the National Natural Science Foundation of China (62173333, 12271522)Beijing Natural Science Foundation (Z210002)the Research Fund of Renmin University of China (2021030187)。
文摘For unachievable tracking problems, where the system output cannot precisely track a given reference, achieving the best possible approximation for the reference trajectory becomes the objective. This study aims to investigate solutions using the Ptype learning control scheme. Initially, we demonstrate the necessity of gradient information for achieving the best approximation.Subsequently, we propose an input-output-driven learning gain design to handle the imprecise gradients of a class of uncertain systems. However, it is discovered that the desired performance may not be attainable when faced with incomplete information.To address this issue, an extended iterative learning control scheme is introduced. In this scheme, the tracking errors are modified through output data sampling, which incorporates lowmemory footprints and offers flexibility in learning gain design.The input sequence is shown to converge towards the desired input, resulting in an output that is closest to the given reference in the least square sense. Numerical simulations are provided to validate the theoretical findings.
基金supported by the National Natural Science Foundation of China(11871312,12131014)the Natural Science Foundation of Shandong Province,China(ZR2023MA086)。
文摘A bicubic B-spline finite element method is proposed to solve optimal control problems governed by fourth-order semilinear parabolic partial differential equations.Its key feature is the selection of bicubic B-splines as trial functions to approximate the state and costate variables in two space dimensions.A Crank-Nicolson difference scheme is constructed for time discretization.The resulting numerical solutions belong to C2in space,and the order of the coefficient matrix is low.Moreover,the Bogner-Fox-Schmit element is considered for comparison.Two numerical experiments demonstrate the feasibility and effectiveness of the proposed method.
文摘In this paper, the optimal control problem of parabolic integro-differential equations is solved by gradient recovery based two-grid finite element method. Piecewise linear functions are used to approximate state and co-state variables, and piecewise constant function is used to approximate control variables. Generally, the optimal conditions for the problem are solved iteratively until the control variable reaches error tolerance. In order to calculate all the variables individually and parallelly, we introduce a gradient recovery based two-grid method. First, we solve the small scaled optimal control problem on coarse grids. Next, we use the gradient recovery technique to recover the gradients of state and co-state variables. Finally, using the recovered variables, we solve the large scaled optimal control problem for all variables independently. Moreover, we estimate priori error for the proposed scheme, and use an example to validate the theoretical results.
文摘In this paper, time-optimal control problem for a liner n× n co-operative parabolic system involving Laplace operator is considered. This problem is, steering an initial state y(0)=u?, with control u?so that an observation y(t) hitting a given target set in minimum time. First, the existence and uniqueness of solutions of such system under conditions on the coefficients are proved. Afterwards necessary and sufficient conditions of optimality are obtained. Finally a scaler case is given.
基金the National Natural Science Foundation of China(No.10674024)
文摘If A： D（A） X→X is a densely defined and closed linear operator, which generates a linear semigroup S （t） in Banach space X. The nonlocal control/ability for the following nonlocal semilinear problems： u＇ （t） = Au （t） ＋ Bx（ t） ＋ f（ t, u（t） ）, 0≤t ≤ T with nonlocal initial condition u（0） = u0 ＋ g（u） is discussed in Banach space X. The results show that if semigroup S（t） is strongly continuous, the functionsf and g are compact and the control B is bounded, then it is nonlocally controllable. The nonlocal controllability for the above nonlocal problem is also studied when B and W are unbounded and the semigroup S（t） is compact or strongly continuous. For illustration, a partial differential equation is worked out.
基金the financial support from the National Natural Science Foundation of China(12171405 and 11661074)the Program for New Century Excellent Talents in Fujian Province University+2 种基金the financial support from the Characteristic&Preponderant Discipline of Key Construction Universities in Zhejiang Province(Zhejiang Gongshang University-Statistics)Collaborative Innovation Center of Statistical Data Engineering Technology&ApplicationDigital+Discipline Construction Project(SZJ2022B004)。
文摘Motivated by recent advances made in the study of dividend control and risk management problems involving the U.S.bankruptcy code,in this paper we follow[44]to revisit the De Finetti dividend control problem under the reorganization process and the regulator's intervention documented in U.S.Chapter 11 bankruptcy.We do this by further accommodating the fixed transaction costs on dividends to imitate the real-world procedure of dividend payments.Incorporating the fixed transaction costs transforms the targeting optimal dividend problem into an impulse control problem rather than a singular control problem,and hence computations and proofs that are distinct from[44]are needed.To account for the financial stress that is due to the more subtle concept of Chapter 11 bankruptcy,the surplus process after dividends is driven by a piece-wise spectrally negative Lévy process with endogenous regime switching.Some explicit expressions of the expected net present values under a double barrier dividend strategy,new to the literature,are established in terms of scale functions.With the help of these expressions,we are able to characterize the optimal strategy among the set of admissible double barrier dividend strategies.When the tail of the Lévy measure is log-convex,this optimal double barrier dividend strategy is then verified as the optimal dividend strategy,solving our optimal impulse control problem.
文摘In the optimal control problem of nonlinear dynamical system,the Hamiltonian formulation is useful and powerful to solve an optimal control force.However,the resulting Euler-Lagrange equations are not easy to solve,when the performance index is complicated,because one may encounter a two-point boundary value problem of nonlinear differential algebraic equations.To be a numerical method,it is hard to exactly preserve all the specified conditions,which might deteriorate the accuracy of numerical solution.With this in mind,we develop a novel algorithm to find the solution of the optimal control problem of nonlinear Duffing oscillator,which can exactly satisfy all the required conditions for the minimality of the performance index.A new idea of shape functions method(SFM)is introduced,from which we can transform the optimal control problems to the initial value problems for the new variables,whose initial values are given arbitrarily,and meanwhile the terminal values are determined iteratively.Numerical examples confirm the high-performance of the iterative algorithms based on the SFM,which are convergence fast,and also provide very accurate solutions.The new algorithm is robust,even large noise is imposed on the input data.
文摘This article presents the Parametric Iteration Method (PIM) for finding optimal control and its corresponding trajectory of linear systems. Without any discretization or transformation, PIM provides a sequence of functions which converges to the exact solution of problem. Our emphasis will be on an auxiliary parameter which directly affects on the rate of convergence. Comparison of PIM and the Variational Iteration Method (VIM) is given to show the preference of PIM over VIM. Numerical results are given for several test examples to demonstrate the applicability and efficiency of the method.
基金Supported by the National Natural Science Foundation of China(62102241)。
文摘Software Defined Network(SDN)has been developed rapidly in technology and popularized in application due to its efficiency and flexibility in network management.In multi-controller SDN architecture,the Controller Placement Problem(CPP)must be solved carefully as it directly affects the whole network performance.This paper proposes a Multi-objective Greedy Optimized K-means Algorithm(MGOKA)to solve this problem to optimize worst-case and average delay between switches and controllers as well as synchronization delay and load balance among controllers for Wide Area Networks(WAN).MGOKA combines the process of network partition based on the K-means algorithm with cluster fusion based on the greedy algorithm and designs a normalization strategy to convert a multi-objective into a single-objective optimization problem.The simulation results depict that in different network scales with different numbers of controllers,the relative optimization rate of our proposed algorithm compared with K-means,K-means++,and GOKA can reach up to 101.5%,109.9%,and 79.8%,respectively.Moreover,the error rate between MGOKA and the global optimal solution is always less than 4%.
文摘In this paper, we consider a fully discrete finite element approximation for time fractional optimal control problems. The state and adjoint state are approximated by triangular linear fi nite elements in space and <em>L</em>1 scheme in time. The control is obtained by the variational discretization technique. The main purpose of this work is to derive the convergence and superconvergence. A numerical example is presented to validate our theoretical results.
基金supported in part by the National Natural Science Foundation of China(62173255,62188101)Shenzhen Key Laboratory of Control Theory and Intelligent Systems(ZDSYS20220330161800001)
文摘Dear Editor,In this letter,a constrained networked predictive control strategy is proposed for the optimal control problem of complex nonlinear highorder fully actuated(HOFA)systems with noises.The method can effectively deal with nonlinearities,constraints,and noises in the system,optimize the performance metric,and present an upper bound on the stable output of the system.
文摘In this paper, we propose the nonconforming virtual element method (NCVEM) discretization for the pointwise control constraint optimal control problem governed by elliptic equations. Based on the NCVEM approximation of state equation and the variational discretization of control variables, we construct a virtual element discrete scheme. For the state, adjoint state and control variable, we obtain the corresponding prior estimate in H<sup>1</sup> and L<sup>2</sup> norms. Finally, some numerical experiments are carried out to support the theoretical results.
文摘Advanced Process Control （APC） is necessary for oil refining and chemical process in China, but some problems have emerged in the application of APC techniques in this field. This paper discusses the conditions of APC application concerning process design, distributed control system （DCS） choice and regular control. It analyzes the problems and strategies in APC application. Some suggestions are proposed for the enterprise to benefit from APC application.
文摘We present in this paper a survey of recent results on the relation between time and norm optimality for linear systems and the infinite dimensional version of Pontryagin＇s maximum principle. In particular, we discuss optimality （or nonoptimality） of singular controls satisfying the maximum principle and smoothness of the costate in function of smoothness of the target.
基金This work was supported by the National Natural Science Foundation of China （Nos. 61227902, 61573343） and the National Center for Mathematics and Interdisciplinary Sciences, CAS.
文摘Fast and high fidelity quantum control is the key technology of quantum computing. The hybrid system composed of the nitrogen-vacancy center and nearby Carbon-13 nuclear spin is expected to solve this problem. The nitrogen-vacancy center electron spin enables fast operations for its strong coupling to the control field, whereas the nuclear spins preserve the coherence for their weak coupling to the environment. In this paper, we describe a strategy to achieve time-optimal control of the Carbon-13 nuclear spin qubit by alternating controlling the nitrogen-vacancy center electron spin as an actuator. We transform the qubit gate operation into a switched system. By using the maximum principle, we study the minimum time control of the switched system and obtain the time-optimal control of the qubit gate operation. We show that the X gate and Y gate operations are within 10μs while the fidelity reaches 0.995.
文摘Iterative methods for solving discrete optimal control problems are constructed and investigated. These discrete problems arise when approximating by finite difference method or by finite element method the optimal control problems which contain a linear elliptic boundary value problem as a state equation, control in the righthand side of the equation or in the boundary conditions, and point-wise constraints for both state and control functions. The convergence of the constructed iterative methods is proved, the implementation problems are discussed, and the numerical comparison of the methods is executed.
基金Project supported by the Key Research and Development Program of Guangdong Province,China(Grant No.2018B030326001)the National Natural Science Foundation of China(Grant Nos.11474152,12074179,U21A20436,and 61521001)the Natural Science Foundation of Jiangsu Province,China(Grant No.BE2021015-1)。
文摘One of the key features required to realize fault-tolerant quantum computation is the robustness of quantum gates against errors.Since geometric quantum gate is naturally insensitivity to noise,it appears to be a promising routine to achieve high-fidelity,robust quantum gates.The implementation of geometric quantum gate however faces some troubles such as its complex interaction among multiple energy levels.Moreover,traditional geometric schemes usually take more time than equivalent dynamical ones.Here,we experimentally demonstrate a geometric gate scheme with the time-optimal control(TOC)technique in a superconducting quantum circuit.With a transmon qubit and operations restricted to two computational levels,we implement a set of geometric gates which exhibit better robustness features against control errors than the dynamical counterparts.The measured fidelities of TOC X gate and X/2 gate are 99.81%and 99.79%respectively.Our work shows a promising routine toward scalable fault-tolerant quantum computation.
文摘A kind of direct methods is presented for the solution of optimal control problems with state constraints. These methods are sequential quadratic programming methods. At every iteration a quadratic programming which is obtained by quadratic approximation to Lagrangian function and linear approximations to constraints is solved to get a search direction for a merit function. The merit function is formulated by augmenting the Lagrangian function with a penalty term. A line search is carried out along the search direction to determine a step length such that the merit function is decreased. The methods presented in this paper include continuous sequential quadratic programming methods and discreate sequential quadratic programming methods.
文摘In this paper, the naturally evolving complex systems, such as biotic and social ones, are considered. Focusing on their structures, a feature is noteworthy, i.e., the similarity in structures. The relations between the functions and behaviors of these systems and their similar structures will be studied. Owing to the management of social systems and the course of evolution of biotic systems may be regarded as control processes, the researches will be within the scope of control problems. Moreover, since it is difficult to model for biotic and social systems, it will start with the control problems of complex systems, possessing similar structures, in engineering. The obtained results show that for either linear or nonlinear systems and for a lot of control problems similar structures lead to a series of simplifications. In general, the original system may be decomposed into reduced amount of subsystems with lower dimensions and simpler structures. By virtue of such subsystems, the control problems of original system can be solved more simply. At last, it turns round to observe the biotic and social systems and some analyses are given.
