Abstract
Full Text
UDC 517.9.004.14
MATHEMATICS
Corresponding Member of the Academy of Sciences of the USSR V. S. VLADIMIROV
ON THE THEORY OF LINEAR PASSIVE SYSTEMS
1. Introduction. By a linear passive system we shall mean a system of convolution equations
\[ Z * u = f, \tag{1} \]
where \(f=(f_1,\ldots,f_N)\) is a given vector distribution from \((\mathcal D')^{\times N}=\mathcal D'\times\cdots\times\mathcal D'\), \(u=(u_1,\ldots,u_N)\) is an unknown element of \((\mathcal D')^{\times N}\), and \(Z=\|Z_{kj}\|\) is an \(N\times N\) matrix with material elements \(Z_{kj}(\xi)\) from \(\mathcal D'=\mathcal D'(\mathbb R^n)\), satisfying the conditions:
a) causality with respect to some convex closed acute cone \(\Gamma\subset \mathbb R^n\) with vertex at \(O\): \(Z(\xi)=0,\ \xi\notin \Gamma\);
b) passivity: for all complex \(a\in \mathbb C^N\), \(\varphi\in\mathcal D\), and \(q\in C=\operatorname{int}\Gamma^*\), the inequality
\[ \operatorname{Re}\int [(Z e^{-(q,\xi)}a,a)*\varphi](x)\,\overline{\varphi}(x)\ge 0; \tag{2} \]
holds; here \(\Gamma^*\) is the cone conjugate to the cone \(\Gamma\).
The operator \(Z*\), defining the linear passive system (1), will be called a passive operator (with respect to the cone \(\Gamma\)), and the corresponding matrix \(Z(\xi)\) the (generalized) impedance of the system.
A number of problems of mathematical physics reduce to linear passive systems: linear thermodynamic systems, the theory of electric circuits, scattering of nuclear particles, scattering of electromagnetic waves, etc. \((^{1-8})\). Inequality (2) reflects the ability of a physical system to absorb energy, but not to generate it.
Linear passive systems have been studied in especially great detail in the case of one independent variable \((^{1-4})\). The principal apparatus of investigation in this case is the classical integral representation of functions holomorphic in the upper half-plane and having there nonnegative real part \((^9)\). Using the results of \((^{10})\) on the representation of holomorphic functions of many complex variables with nonnegative real part in a tubular domain over a cone, we have investigated linear passive systems in the multidimensional case. Here we present the most important results of these investigations.
2. Characteristic of a passive operator. From conditions a) and b) it follows that \(Z_{kj}\in \mathcal S'(\Gamma)\), so that the matrix \(Z(\xi)\) has a Fourier–Laplace transform \((^{11})\).
Theorem 1. In order that an \(N\times N\) matrix \(Z(\xi)\) define a passive operator with respect to the cone \(\Gamma\), it is necessary and sufficient that its Fourier–Laplace transform \(\widetilde Z(\zeta)\), \(\zeta=p+iq\), be a holomorphic matrix-function in the tubular domain \(T^C=\mathbb R^n+iC,\ C=\operatorname{int}\Gamma^*\), satisfying the relation
\[ \widetilde Z(p+iq)=\overline{\widetilde Z(-p+iq)},\qquad \zeta\in T^C, \tag{3} \]
and the inequality
\[ \operatorname{Re}(\widetilde Z(\zeta)a,a)\ge 0,\qquad a\in\mathbb C^N,\quad \zeta\in T^C. \tag{4} \]
Remark. The sufficient conditions in Theorem 1 admit a weakening: if the matrix \(\widetilde Z(\zeta)\) is holomorphic in the tube domain \(T^C\) (the cone \(C\) may also be nonconvex), satisfies inequality (4) in \(T^C\), and the matrix \(\widetilde Z(iq)\) is real for all \(q\in C\), then the matrix \(Z(\xi)\) defines a passive operator with respect to the cone \(C^*\).
Theorem 2. In order that an \(N\times N\)-matrix \(Z(\xi)\) define a passive operator with respect to the cone \(\Gamma\), it is necessary and sufficient that, for any cone
\[ C'=[q:\ (e_j,q)>0,\ j=1,\ldots,n]\subset C=\operatorname{int}\Gamma^* \]
it be representable (uniquely) in the form \(^*\)
\[ Z(\xi)=(e_1,D)^2\ldots(e_n,D)^2 Z_{C'}(\xi)+\sum_{j=1}^n Z_{C'}^{(j)}D_j\delta(\xi), \tag{5} \]
where \(Z_{C'}(\xi)\) is a real continuous matrix of polynomial growth in \(\mathbb R^n\) with support in the cone \((C')^*\), and such that for any \(a\in \mathbb C^N\) the generalized function \(^{**}\)
\[ (e_1,D)^2\ldots(e_n,D)^2\bigl([Z_{C'}(\xi)+Z_{C'}^T(-\xi)]a,a\bigr) \]
is positive-definite in the sense of Bochner–Schwartz; the matrices \(Z_{C'}^{(j)}\), \(j=1,\ldots,n\), are real symmetric and such that the matrix
\[ \sum_{j=1}^n q_j Z_{C'}^{(j)}\geqslant 0 \]
for all \(q\in C'\).
Remark. For \(n=1\), Theorem 2 was proved by König and Zemanian \((^4)\).
Corollary. In order that a function \(f(\zeta)\) be holomorphic in \(T^C\), \(\operatorname{Im} f(\zeta)\geqslant 0\), \(\zeta\in T^C\), and
\[ \lim \operatorname{Im} f(p+iq)=0,\qquad q\to 0,\qquad q\in C \text{ in } \mathscr S', \]
it is necessary and sufficient that it be representable in the form \(f(\zeta)=(a,\zeta)\), where \(a\in C^*\) (a theorem of the type of the Liouville and Phragmén–Lindelöf theorems).
3. Multidimensional dispersion relations
Theorem 3. In order that an \(N\times N\)-matrix \(Z(\xi)\) define a passive operator with respect to the cone \(\Gamma\), it is necessary and sufficient that its Fourier transform \(\widetilde Z(p)\), for any cone \(C'\subset C=\operatorname{int}\Gamma^*\) (see Theorem 2), satisfy the dispersion relation
\[ \widetilde Z(p)= \frac{2(-1)^n}{(2\pi)^n}(e_1,p)^2\ldots(e_n,p)^2(\widetilde M_{C'}*K_{C'}) -i\sum_{j=1}^n Z_{C'}^{(j)}p_j; \tag{6} \]
where
\[ K_{C'}(p)=\det\|e_{jk}\|\left\{-i\pi\delta[(e_1,p)]+P\frac{1}{(e_1,p)}\right\}\times\ldots \]
\[ \ldots\times\left\{-i\pi\delta[(e_n,p)]+P\frac{1}{(e_n,p)}\right\}; \]
\(\widetilde M_{C'}(p)=F[M_{C'}(\xi)]\) is a Hermitian matrix with the properties: I. The matrix \(M_{C'}(\xi)\) is real continuous of polynomial growth with support in the cone \((C')^*\cup(-C')^*\). II. For any \(a\in\mathbb C^N\) the generalized function
\[ \frac{(-1)^n(e_1,p)^2\ldots(e_n,p)^2} {[1+(e_1,p)^2]\ldots[1+(e_n,p)^2]} (\widetilde M_{C'}(p)a,a) \]
is a nonnegative finite measure on \(\mathbb R^n\); \(Z_{C'}^{(j)}\), \(j=1,\ldots,n\), are real symmetric matrices such that for all \(q\in C'\)
\(^*\) \(D=(D_1,\ldots,D_n)=(\partial/\partial x_1,\ldots,\partial/\partial x_n)\).
\(^ {**}\) \(Z^T\) is the matrix transposed to the matrix \(Z\).
\[ \sum_{j=1}^{n} q_j Z_C^{(j)} \geqslant 0. \]
In this case the equality holds
\[ M_{C'}(\xi)=\frac12\left[Z_{C'}(\xi)+Z_{C'}^{T}(-\xi)\right]. \tag{7} \]
Remark. For \(n=1\), Theorem 3 was proved by Beltrami and Wohlers \({}^{(3)}\).
4. Existence of a fundamental solution. A fundamental solution of the passive operator \(Z*\) is any matrix \(A(\xi)\), \(A_{kj}\in\mathcal D'\), satisfying the matrix convolution equation
\[ Z*A=I\delta(\xi). \tag{8} \]
The operator \(A*\) is also called the inverse operator to the operator \(Z*\), and the matrix \(A(\xi)\) the (generalized) admittance of the system. A passive operator \(Z*\) is called nondegenerate (completely nondegenerate) if \(\det Z(\zeta)\ne0\), \(\zeta\in T^C\) (respectively, for every \(a\in C^N\), \(a\ne0\), there exists a point \(\zeta_0\in T^C\) such that \(\operatorname{Re}(Z(\zeta_0)a,a)>0\)).
In order that the passive operator \(Z*\) be completely nondegenerate, it is necessary and sufficient that
\[ (Z(\xi)a,a)\ne i g^0(\xi),\qquad a\in C^N,\quad a\ne0,\quad \operatorname{Im} g=0. \tag{9} \]
The following is the main result.
Theorem 4. In order that the operator \(Z*\), \(Z_{kj}\in\mathcal D'(\Gamma)\), be a (completely) nondegenerate passive operator with respect to the cone \(\Gamma\), it is necessary and sufficient that there exist a unique inverse operator \(A*\) in the class of (completely) nondegenerate passive operators with respect to \(\Gamma\).
5. Examples of passive systems. We first note a sufficient condition for passivity. Let a real \(N\times N\) matrix \(Z(\xi)\) satisfy the causality condition a), item 1, and the condition
\[ \text{b′)}\quad \operatorname{Re}\int_{-\Gamma}\left[((Za,a)*\varphi](x)\,\overline{\varphi}(x)\right]\,dx\geqslant0, \qquad \varphi\in\mathcal D,\quad a\in C^N. \tag{10} \]
(We assume that the cone \(\Gamma\) satisfies the conditions of item 1 and, moreover, is solid.) Then the matrix \(Z(\xi)\) defines a passive operator with respect to the cone \(\Gamma\), i.e., it satisfies condition b), item 1.
1) If a system of differential equations with constant coefficients is passive, then it is of first order.
2) In order that the differential operator \([(l,D)+c]\delta(\xi)*\) with constant coefficients be passive with respect to the cone \(\Gamma\), it is necessary and sufficient that \(l\in\Gamma\) and \(c\geqslant0\).
3) In order that a system of linear differential equations with real constant coefficients
\[ \sum_{k=1}^{n} A_k \frac{\partial u}{\partial x_k}=f, \tag{11} \]
be passive and completely nondegenerate, it is necessary and sufficient that the \(N\times N\) matrices \(A_k\), \(k=1,\ldots,n\), be symmetric and that there exist such a vector \(l\) that
\[ \sum_{k=1}^{n} l_k A_k>0. \]
In this case the passivity of the system (11) holds with respect to the convex hull of the cone
\[ [\xi:\ \xi_1=(A_1a,a),\ldots,\xi_n=(A_na,a),\quad a\in R^N]. \]
Remark. Systems (11) satisfying the conditions of assertion 3) are the principal parts of systems symmetric in the sense of Friedrichs \({}^{(13)}\) with constant coefficients.
4) In order that the difference operator
\[ B_{0}\delta(\xi)_{*}+\sum_{\nu=1}^{m}B_{\nu}\delta(\xi-h_{\nu})_{*}, \]
where \(B_{k}\), \(k=0,1,\ldots,m\), are real \(N\times N\) matrices, be passive, it is necessary and sufficient that: I. The smallest closed convex cone \(\Gamma\) containing the points \(\{0,h_{\nu},\nu=1,\ldots,m\}\) be acute. II. For all \(\zeta=p+iq\in T^{C}\), \(C=\operatorname{int}\Gamma^{*}\), the matrix
\[ B_{0}+B_{0}^{T}+\sum_{\nu=1}^{m}e^{-(q,h_{\nu})} \left[ \cos(p,h_{\nu})(B_{\nu}+B_{\nu}^{T}) -\sin(p,h_{\nu})\frac{B_{\nu}-B_{\nu}^{T}}{i} \right]\geqslant 0 . \]
In this case passivity holds with respect to the cone \(\Gamma\).
Correction note. It can be proved that every matrix \(Z(\xi)\) defining a passive operator with respect to a solid cone \(\Gamma\) also satisfies condition \(b'\).
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
13 III 1969
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