Partial linear integrals of the Poincaré–Zhukovskii equations (the general case)

Linear invariant relations of Kirchhoff's equations

A new linear invariant relation of the Poincaré–Zhukovskii equations

Linear invariant relations of the Poincaré–Zhukovskii equations

For the Kirchhoff equations in the general case, when the cross-term matrix of the Hamiltonian can be asymmetric, the existence conditions for a linear invariant relation of a general form that connect the impulsive moment and the impulsive force are obtained. It is shown that equations with such an invariant relation can be transformed into the equations with an invariant relation for an impulsive moment, the existence conditions of which with a symmetric cross-term matrix coincide with the Chaplygin case. A description of a set of Hamiltonians that admit the existence of a linear invariant relation of a general form is given. The coordinate form of the invariant relation and its existence conditions is obtained. The total number of conditions is six (in contrast to the eight conditions in the Chaplygin case). Reduction to the Riccati equation is preformed. It is shown that if there is a linear integral, then the Kirchhoff equations are reduced to the Kirchhoff case.

A linear invariant relation in the problem of the motion of a gyrostat in a magnetic field

Some invariant relations in the problem of the motion of a body on a smooth horizontal plane

The integration of Poisson's equations in the case of three linear invariant relations

The problem of the free motion of two bodies connected by a pair of spherical hinges is considered. The conditions for the existence of invariant relations analogous to the Hess integral are specified.

On certain linear invariant relations between the elements of a square matrix

Consider the minimization of the following quadratic cost functional: $$J(u):=E\langle Mx_T,x_T\rangle +E\int_0^T(\langle Q_sx_s,x_s\rangle +\langle N_su_s,u_s\rangle )\, ds,$$ where x is the solution of the following linear stochastic control system: $$ \eq{dx_t=&(A_tx_t+B_tu_t)\, dt +\sum_{i=1}^d(C_t^ix_t+D_t^iu_t)\, dW_t^i,\cr x_0=&h\in \mathbb{R}^n,\qquad u_t\in \mathbb{R}^m; \cr} $$ u is a square integrable adapted process. The problem is conventionally called the stochastic LQ (the abbreviation of "linear quadratic") problem. We are concerned with the following general case: the coefficients A,B,Ci,Di, Q, N, and M are allowed to be adapted processes or random matrices. We prove the existence and uniqueness result for the associated Riccati equation, which in our general case is a backward stochastic differential equation with the generator (the drift term) being highly nonlinear in the two unknown variables. This solves Bismut and Peng's long-standing open problem (for the case of a Brownian filtration), which was initially proposed by the French mathematician J. M. Bismut [in Séminaire de Probabilités XII, Lecture Notes in Math. 649, C. Dellacherie, P. A. Meyer, and M. Weil, eds., Springer-Verlag, Berlin, 1978, pp. 180--264]. We also provide a rigorous derivation of the Riccati equation from the stochastic Hamilton system. This completes the interrelationship between the Riccati equation and the stochastic Hamilton system as two different but equivalent tools for the stochastic LQ problem. There are two key points in our arguments. The first one is to connect the existence of the solution of the Riccati equation to the homomorphism of the stochastic flows derived from the optimally controlled system. Actually, we establish their equivalence. As a consequence, we can construct solutions to a sequence of suitably modified Riccati equations in terms of the associated stochastic Hamilton systems (and the optimal controls). The second key point is to establish a new type of a priori estimate for solutions of Riccati equations, with which we show that the sequence of constructed solutions has a limit which is a solution to the original Riccati equation.

Matrix quadratic equation solution derivation applied in finding steady state solutions of Riccati differential equations with constant coefficients

Periodic properties of solutions play an important role in characterizing the behavior of solutions of sufficiently complicated nonlinear differential equations. Sufficient conditions are established which ensure the existence of periodic (or almost periodic) solutions of certain second nonlinear differential equations. Using the basic tool Lyapunov function, new result on the subject which improve some well known results in the literature with the particular cases of (1) for the existence of almost periodic or periodic solutions when the forcing term $p$ is almost periodic or periodic in t uniformly in $x$ and $\dot{x}$ are obtained. Our result further extends and improves on those that exist in the literature to the more general case considered.

The method of Mironenko’s reflecting function is used for investigation of Riccati equations. The class of Riccati equations with certain-type reflecting function has been preliminarily constructed. The necessary and sufficient conditions, under which the Riccati equation would have a reflecting function linear in phase variable, are proved. These conditions are constructive in nature, since on their basis the formula is obtained, which shows the linear in phase variable reflecting function in terms of the coefficients of the Riccati equation. Additionally, the relationship between the parity (oddness) property of the coefficients of the Riccati equation and the existence of a reflecting function linear in phase variable is investigated. The application of the method of Mironenko’s reflecting function to the constructed class of Riccati equations revealed sufficient conditions, under which all its solutions are periodic or almost periodic. A sign of no periodic solutions for almost periodic Riccati equations is obtained. An example of the quasi-periodic Riccati equation with quasi-periodic reflecting function, which has a periodic solution, is given.

T HE relative motion of a follower satellite with respect to the leader in a given circular orbit is described by autonomous nonlinear differential equations. The linearized equations around the null solution are known as Hill–Clohessy–Wiltshire (HCW) equations [1–3]. The HCWequations were used by many authors to study rendezvous problems (see [4] and references therein). The Tschauner–Hempel (TH) equations replace the HCW equations, when the orbit of the leader is eccentric [2,5]. Rendezvous problems along an eccentric orbit were studied in [4,5]. The HCW equations possess periodic solutions, which are useful as temporary orbits before mission and for proximity operations such as inspection and repair. The leader–follower formation and reconfiguration problems based on the periodic solutions were studied bymany authors [6–10]. The TH equations also have periodic solutions. They are characterized by Inalhan et al. [11], and the initialization procedure to periodic motion is given. Periodic solutions also follow from the transition matrix of the TH system given byYamanaka and Ankersen [12]. The effects of eccentricity on the shape and size of relative orbits are studied by Sengupta andVadali [13]. Periodic solutions of the TH equations are used for formation flying [14,15] because no control efforts are needed to maintain them. However, their period is fixed and is equal to that of the leader orbit, which would be inconvenient for a quick inspection of the leader. The shape of periodic solutions is irregular compared with that of the HCWequations, which would be undesirable for some missions. In this Note, active formation flying for the TH system is considered, in which the desired relative orbit of the follower is generated by an exosystem. This allows for flexibility of the shape and period of the reference orbit. Typical examples are elliptic relative orbits of the HCWequations with higher frequencies. Formation flying for the TH system with generated reference orbits has not been studied in the literature. To realize such a formation flying, the output regulation theory for linear periodic systems recently given by Ichikawa and Katayama [16] is employed. Output regulation covers tracking and disturbance rejection [17], but the tracking aspect of the theory is used. The regulator equation, which is necessary to achieve output regulation, is a differential equationwith periodic coefficients. The main contribution of this Note is to show that the regulator equation can be solved algebraically if the system is in the controllable canonical form and the observation matrix is of a special form. To achieve asymptotic tracking, stabilizing feedback controls are necessary. They are designed by the differential Riccati equation (DRE) of the linear quadratic regulator theory [18]. To show the effectiveness of this approach, two examples are given. In the first example, the orbit of the leader is assumed to be circular and the HCW equations are considered. The reference orbit of the follower for formation flying is circularwith arbitrary frequency. Starting from a periodic solution of the HCW equations, the follower satellite is asymptotically steered to the reference orbit. In this case, both the regulator equation and theRiccati equation become algebraic. TheL1 norm of the input ΔV and the settling time are given as functions of the weight parameter in the Riccati equation. The L1 norm of the input to maintain the reference orbit for one period is also calculated. These performance indices are further examined by varying the frequency of the reference orbit. The second example is concerned with the TH equations for the elliptic leader orbit. The reference relative orbit of the follower is circular, as in the first example. The regulator equation has periodic coefficients, but it is algebraically solved and explicitly given. The DRE is solved backward and the stabilizing periodic solution is obtained. TheL1 normof the input and the settling time are computed by varying eccentricity and frequency of the reference orbit.

A scheme for a low-voltage CMOS syllabic-companding log domain filter with wide dynamic range is proposed and its prototype is presented. A nodal voltage which is fixed in a conventional filter based on the dynamically adjustable biasing (DAB) technique is adapted for change of input envelope to achieve wide dynamic range. Externally linear and time invariant (ELTI) relation between an input and an output is guaranteed by a state variable correction (SVC) circuit which is also proposed for low-voltage operation. To demonstrate the proposed scheme, a fifth-order Chebychev low-pass filter with 100-kHz cutoff frequency is designed and fabricated in a standard 0.35-μm CMOS process. The filter has a 78-dB dynamic range and consumes 200-μW power from a 0.8-V power supply.