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Conventionally, the Three-Loop Autopilot is dominated by three main parameters namely the damping factor, time constant and the open-loop crossover frequency, to achieve the desired performance. In developing the autopilot gains, the same gains as the conventional approach can also be obtained using the general optimality theory. The basic idea of setting the design parameters and the feedback gains using methods of optimal control revolves around the proper selection of the performance index for which the control is optimized. In this paper, an explicit formula for autopilot gains is driven by taking into account the relations between the desired performance and the open-loop frequency, phase margin and time constant without the need to adjust the LQR weights. Since each set of design parameters selection doesn’t guarantee the existence of the optimal control law, the optimization criteria for the three-loop autopilot are derived to know the set of design parameters for which the control is optimized. Finally, an optimal criterion to select the design parameters and therefore the autopilot gains is developed, where the time constant is set to the designer objective while the open-loop crossover frequency and the phase margin as design constraints.

Based on the computerized symbolic system Maple and a Riccati equation, a new Riccati equation expansion method for constructing nontraveling wave and coefficient functions' soliton-like solutions is presented by a new general ansätz. The proposed method is more powerful than most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method, and generalized hyperbolic-function method. By using the method, we not only successfully recovered the previously known formal solutions but could also construct new and more general formal solutions for some nonlinear differential equations. Making use of the method, we study the (2+1)-dimensional Boussinesq equation and obtain rich new families of the exact solutions, including the nontraveling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, and triangular functions solutions.

In this paper, a generalized extended tanh-function method for constructing exact solutions of nonlinear evolution equations (NEEs) is presented and implemented in a computer algebraic system. Applying the generalized method, with the aid of Maple, we consider the system of the approximate equations for long water waves and find the new more general explicit exact solutions. The properties of the new solitary wave solutions are shown by some figures.

The extended simple equation method is applied to construct solitary wave solutions of (3 + 1)-dimensional Kadomtsev–Petviashvili-Benjamin–Bona–Mahony (KP-BBM), Korteweg–de Vries Benjamin–Bona–Mahony (KdV-BBM), Breaking soliton (BS) and (2 + 1) Maccari system waves system of equations. These models have prevalent usage in modern science. This technique can also be functional to solve different kinds of nonlinear evolution problems in contemporary areas of research. It is an effective and powerful mathematical tool in finding solitary wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics.

The factorization method of the Hamiltonian in quantum mechanics is used to solve a particular type of Riccati equation. In this paper, He's homotopy perturbation method is applied to a general Riccati equation. The obtained solutions are in recursive sequence forms and, in many cases, these solutions lead to an exact solution. This property generally means that one can solve completely the eigenvalue problem for the Hamiltonian operator. The method is tested on various examples which are revealing the effectiveness and the simplicity of the method.

Based on the rational solutions to a generalized Riccati equation, a new method which is called as rational function method is proposed. We apply this method to solve the coupled mKdV equations and derive a set of rational solutions. This method is also available for seeking their solutions to other NLPDEs. It shows that the rational function method is more universal and powerful than the auxiliary Riccati equation method.

According to two dependent solutions to a generalized Riccati equation together with the equation itself, a multiple Riccati equations rational-exponent method is proposed and applied to Whitham–Broer–Kaup equation. It shows that this method is a more concise and efficient approach and can uniformly derive many types of combined solutions to nonlinear partial differential equations.

In this research paper, we take into account the ($2+1$)-dimensional Kadomtsev–Petviashvili equation which is important in the soliton theory of nonlinear physics. To get the desired soliton solutions, the modified F-expansion method using the Riccati equation which has many solution functions, as well as the modified generalized Kudryashov’s method, had been effectively implemented. One of the reasons for the preference of the methods is that the proposed methods have been widely used before and they have not been applied to this problem. First, the wave transform is applied to the considered nonlinear partial differential equation (NLPDE), the nonlinear ordinary differential equation (NODE) form and the balancing constant are determined. The next step is to use the auxiliary equation depending on the proposed method to find the solution of the NODE form and to obtain the linear algebraic equation system. The solution of this system gives different solution sets for unknown parameter values. Then, soliton solution functions are constructed by using the suitable solution sets. After testing and confirming that the obtained solution functions satisfy the main equation, the three- and two-dimensional illustrations are depicted.

Based on a Bäcklund transformation of the Riccati equation and its known soliton solutions, we obtain in this paper some exact traveling-wave solutions, including triangle function solutions and hyperbolic function solutions, of a hybrid lattice equation. The proposed method can be easily extended to locate exact solitary wave solutions for other types of differential-difference equations.

A new Lax pair is introduced for a perturbed Kaup–Newell equation and used to construct two series of conservation laws through a Riccati equation that a ratio of eigenfunctions satisfies. Both series of conservation laws are defined recursively, and the first two in each series are presented explicitly.

In this paper, new traveling wave solutions of time fractional Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation have been revealed by the help of extended Kudryashov’s method. In this case, the time fractional derivative has been considered in the sense of conformable fractional derivative. The salient feature of the proposed extended Kudryashov’s method is to take full advantage of solutions for the Riccati and the Bernoulli equations involving parameters. Using some particular wave transformation, the given equation is converted to the nonlinear ODE by the help of chain rule and the derivative of a composite function; and then, a novel analytical method viz. extended Kudryashov’s method has been used to solve the resulting equation. Consequently, some new exact solutions have been successfully obtained. The exact solutions devised by the proposed method manifest that the proposed method is effective and easy to implement.

In this paper, the new exact solutions have been studied for conformable time-fractional Rosenau–Kawahara-RLW (RK-RLW) equation. One of the most relevant and precise analytical methods has been employed here for generating new exact solutions of conformable time-fractional RK-RLW equation. Also, the properties of conformable fractional derivative have been used for reduction of conformable time-fractional RK-RLW equation. Furthermore, the nature of the solutions with physical explanation has been presented.

This paper presents new optical solitons of a fractal Fokas–Lenells equation that models the dynamics of optical fibers. The Painlevé technique is employed to identify kink soliton solutions. The constraint conditions guarantee the existence of these solitons. The outcomes of this research give new solutions that are not achieved using some already defined algorithms. The derived method is efficient and its applications are promising for other nonlinear problems. The 3D graphical illustrations of obtained results are depicted for various values of the fractal parameter.

In this paper, we examined four different forms of generalized (2+1)-dimensional Boussinesq–Kadomtsev–Petviashvili (B-KP)-like equations. In this connection, an accurate computational method based on the Riccati equation called sub-equation method and its Bäcklund transformation is employed. Using this method, numerous exact solutions that do not exist in the literature have been obtained in the form of trigonometric, hyperbolic, and rational. These solutions are of considerable importance in applied sciences, coastal, and ocean engineering, where the B–KP-like equations modeled for some significant physical phenomenon. The graph of the bright and dark solitons is presented in order to demonstrate the influence of different physical parameters on the solutions. All of the findings prove the stability, effectiveness, and accuracy of the proposed method.

In this paper, we consider the Boussinesq equation which is an important equation and it is widely used in coastal engineering, harbors, shallow seas and water wave to model weakly nonlinear and long wave approximation. Exact traveling wave solutions in such equations are extremely valuable in analytical and numerical theories. To compute the soliton solutions of this model, we employ an enhanced modified extended tanh expansion method. We demonstrate that the characteristics of soliton solutions may reflect the dispersion of propagation on wave fronts and have an acceptable reliance on parameter values. Bright, dark, bright-singular, dark-singular, periodic-bright singular, periodic dark-singular solutions identified and three-dimensional (3D), contour and two-dimensional (2D) layouts are represented graphically.

This study purposes to extract some fractional analytical solutions of the Peyrard-Bishop-Deoxyribo-Nucleic-Acid ($\beta$-PBDNA) dynamic model with the beta-derivative by the unified Riccati equation expansion method (UREEM). Furthermore, we examine the role of various parameters of the fractional model on the soliton dynamic. The research focuses on computational biophysics and materials science, examining the impact of various parameters on the fractional model. This paper contributes to understanding soliton solutions and the $\beta$-PBDNA dynamic model, demonstrating the applicability of the UREEM method to various fractional models. Some soliton solutions of the model are successfully generated by applying the UREEM. Implementing the UREEM, we take a fractional wave transformation to convert the model into a nonlinear ordinary differential equation. So, a linear equation system is generated. After the system is solved, the soliton solutions are gained for the appropriate solution sets. Finally, 3D, 2D and contour graphs of diverse solutions are depicted at suitable values of parameters. In addition, this paper presents 3D, 2D and contour graphs of various solutions with suitable parameter values. The results are beneficial for interpreting the model in future work and confirm that UREEM is effectively applicable to diverse fractional models, coupled with a comprehensive graphical analysis of how different parameters influence these solutions.

This paper presents a simple and systematic approach to the design of an H_∞ loop-shaping controller for a boost converter operating in continuous conduction mode. Using an averaged linear model of the converter, a robust controller is developed, guaranteeing stability, and the desired closed-loop dynamic response. A controller exists if and only if the unique stabilizing solutions to two algebric Riccati equations are positive definite and the spectral radius of their product is less than . Under these conditions, a compact control structure is obtained and the controller is parameterized. The main contribution of the tuning rules is that the controller parameters can be obtained using one design parameter which reflects the trade-off between stability robustness and time domain performance. Various aspects of the design, including the associated practical problems and the proposed solutions, are detailed. Simulation results are presented to illustrate the design procedure.

Gravitationally coupled scalar fields ϕ, distinguished by the choice of an effective self-interaction potential V(ϕ), simulating a temporarily nonvanishing cosmological term, can generate both inflation and late time acceleration. In scalar field cosmological models the evolution of the Hubble function is determined, in terms of the interaction potential, by a Riccati type equation. In the present work, we investigate scalar field cosmological models that can be obtained as solutions of the Riccati evolution equation for the Hubble function. Four exact integrability cases of the field equations are presented, representing classes of general solutions of the Riccati evolution equation. The solutions correspond to cosmological models in which the Hubble function is proportional to the scalar field potential plus a linearly decreasing function of time, models with the time variation of the scalar field potential proportional to the potential minus its square, models in which the potential is the sum of an arbitrary function and the square of the function integral, and models in which the potential is the sum of an arbitrary function and the derivative of its square root, respectively. The cosmological properties of all models are investigated in detail, and it is shown that they can describe the inflationary or the late accelerating phase in the evolution of the universe.

Exact solutions of the fractional Kraenkel–Manna–Merle system in saturated ferromagnetic materials have been studied. Using the fractional complex transforms, the fractional Kraenkel–Manna–Merle system is reduced to ordinary differential equations, (1 + 1)-dimensional partial differential equations and (2 + 1)-dimensional partial differential equations. Based on the obtained ordinary differential equations and taking advantage of the available solutions of Jacobi elliptic equation and Riccati equation, soliton solutions, combined soliton solutions, combined Jacobi elliptic function solutions, triangular periodic solutions and rational function solutions, for the KMM system are obtained. For the obtained (1 + 1)-dimensional partial differential equations, we get the classification of Lie symmetries. Starting from a Lie symmetry, we get a symmetry reduction equation. Solving the symmetry reduction equation by the power series method, power series solutions for the KMM system are obtained. For the obtained (2 + 1)-dimensional partial differential equations, we derive their bilinear form and two-soliton solution. The bilinear form can also be used to study the lump solutions, rogue wave solutions and breathing wave solutions.

An optimal investment problem is considered for a continuous-time market consisting of the usual bank account, a rolling horizon bond, and a discount bond whose maturity coincides with the planning horizon. Two economic factors, namely, the short rate and the risk-free yield of some fixed maturity, are modeled as Gaussian processes. For the problem of maximizing expected CRRA utility of terminal wealth, the optimal portfolio is obtained through a Bellman equation. The results are noteworthy because the discount bond, which is the riskless asset for the investor, causes a degeneracy due to its zero volatility at the planning horizon. Indeed, this delicate matter is treated rigorously for what seems to be the first time, and it is shown that there exists an optimal, admissible (but unbounded) trading strategy.