【正文】 view the role of the matrix exponential, mention a few of its interesting properties, and point out some related unsolved problems. Consider the linear systemwhere the state , the control and .The state x(t) is given explicitly by the wellknown formula If x0= 0 and u(.) is allowed to be an arbitrary integrable function on the interval [0, t), then the set of all states x(t) reachable at time t is the subspace of Rn given by the range of the nonnegative definite matrix defined byFurthermore, for t 0 the range of Q(t) is independent of t and is given by [18, 56, 74, 951]where Im denotes image or range. If A is asymptotically stable, then exists and is given by the controllability Gramian which is the unique solution to the Lyapunov equation [53, 56, 74, 951]Interesting problems arise immediately if the matrix A is perturbed by another matrix, say A’. For example, it may be of interest to understand the relationship between eA and eA+A’ (where t = 1 for convenience here). If A and A’ mute, then clearly eA+A’= eAeA’= eAeA, whereas if A and A’ do not mute then eA+A’, eAeA, and eAeA are generally different [13]. Furthermore,as shown by examples in [85], eAeA’= eAeA does not imply eAeA’= eA+A’, eA+A’= eAeA’= eAeA does not imply AA’= A’A ,and eAeA’=eA+A’, and eAeA’= however, A and A’ have only algebraic entries then eAeA’= eAeA implies that A and A’ mute [85, 86]. If A and A’have algebraic entries and eAeA’= eA+A’, then it is reasonable to conjecture that A and A’ must mute, but this case is not discussed in [85] and remains open. Specializing to the case A’ = AT, a related open question is the following [14]: Does there exist a nonnormal matrix A satisfying either Some relevant results are given in [75]. In a somewhat different vein, the CampbellBakerHausdorff formula from Lie group theory [l0, 79, 83, 84, 87] states that if A and A’ have sufficiently small norm, then there exists a matrix D in the Lie algebra generated by { A, A’} that satisfies Specifically, D is given by the expansion where . Of course, at least one such matrix D satisfying () must always exist, and it need not be unique . The expansion (), however, is only locally convergent [83]. Thus () can only be used to determine the existence of D in the Lie algebra generated by { A, A’} when the norm of [A, A’] is sufficiently small.A remarkable result of a related, but slightly different, nature is given in [80]. If A and A’ have sufficiently small norm, then there exist invertible matrices S and T (depending upon A and A’) such that Furthermore, it is known that S and T are of the form ep and eQ where P and Q are elements of the Lie algebra generated by A and A’. An alternative, globally convergent expansion is given by [70] where, for k = 0, 1,…， Another class of related results involves inequalities for spectral functions of products of exponentials. Such bounds may be useful for robust stability of sampleddata control systems [16]. For example, if A and A’ are symmetric, then [24, 57]while for arbitrary A we also have [12, 23]A closely related result is An interesting open question that immediately arises is whether or not it is possible to derive ()()directly from any of the formulas (), () ,or (). In this regard () appears to be the most promising candidate. Finally, note that for implies Hence if A is stable, then the lefthand side of ()will converge to zero, whereas the righthand side may beunbounded, rendering the bound useless. A generalization of () in the spirit of () with A’= AT may be useful here. To resolve the possible conservatism in () for it is natural to conjecture generalizations of () to include terms of the form , where the positive definite matrix P is chosen as in Lyapunov stability theory to render ATP + PA negative definite. 翻譯：在線性系統(tǒng)和控制中出現(xiàn)的矩陣?yán)碚撻_放性問題摘要控制理論一直以來都提供了矩陣?yán)碚摪l(fā)展的強(qiáng)大動(dòng)力。因此,我們討論在矩陣?yán)碚摰囊恍╅_放性問題從理論和實(shí)踐的問題出現(xiàn)在線性系統(tǒng)理論及反饋控制。討論了魯棒穩(wěn)定性的問題,包括規(guī)格、穩(wěn)定exponentamp。誘導(dǎo)矩陣可轉(zhuǎn)讓性產(chǎn)生與桿,和非標(biāo)準(zhǔn)的矩陣方程。大量的文獻(xiàn)包括來了解矩陣?yán)碚搶W(xué)家與存在的問題,并應(yīng)用領(lǐng)域的發(fā)展趨勢(shì)。反饋控制理論長(zhǎng)期提供一個(gè)資源豐富的動(dòng)機(jī)在矩陣?yán)碚摰陌l(fā)展。本文的目的是討論幾個(gè)有待解決的問題在矩陣?yán)碚?從理論和實(shí)踐的問題出現(xiàn)在反饋控制理論和相關(guān)地區(qū)的線性系統(tǒng)理論。許多的這些問題都出奇的簡(jiǎn)單,具有強(qiáng)烈的興趣,國(guó)家在控制理論與應(yīng)用,卻仍是一個(gè)未解決的問題。除了導(dǎo)致這些問題的解決,所以我們希望本文有助于刺激增長(zhǎng)之間的交互作用矩陣和控制理論家。因此,本文主要包括一些簡(jiǎn)單教程中討論這些問題,并提供動(dòng)機(jī)。我們討論的問題被分成5個(gè)主題,即魯棒穩(wěn)定性、矩exponentials、誘導(dǎo)規(guī)范、穩(wěn)定與桿可轉(zhuǎn)讓性產(chǎn)生,和非標(biāo)準(zhǔn)的矩陣方程。值得注意的是,這些問題不屬于我自己,但是源于不同種類的控制和matrixtheory應(yīng)用場(chǎng)合,由于大量的研究人員。2. 魯棒穩(wěn)定性一個(gè)基礎(chǔ)性的問題分析的線性系統(tǒng)如下: 給定一個(gè)集矩陣，確定一個(gè)子集。這樣,如果的每個(gè)元素都是穩(wěn)定的，（也就是說,它的各個(gè)特征值負(fù)實(shí)際的部分），然后每個(gè)元素的也是穩(wěn)定的。這個(gè)問題出現(xiàn)時(shí),是不確定建模數(shù)據(jù),保證穩(wěn)定的需要。一個(gè)相關(guān)的問題涉及到一組多項(xiàng)式而不是一套矩陣。例如,考慮多項(xiàng)式的集合：在這里，從i=0到i=n1，系數(shù)β的上下限是給定的，在這種情況下,相當(dāng)顯著的成效是的Kharitonov陳述的的每個(gè)元素都是穩(wěn)定的,如果的每一個(gè)元素是穩(wěn)定的, 這里 是 的子集，它由以下四種多項(xiàng)式組成的:隨著s的減小，系數(shù)的4cyclic圖案的重復(fù)。因此,來決定是否在 的每一個(gè)多項(xiàng)式是穩(wěn)定的，只檢查這四個(gè)多項(xiàng)式就足夠了。Kharitonov的結(jié)果已經(jīng)引起了相當(dāng)大的興趣并在眾多方向推廣[6 7 9].矩陣相應(yīng)的問題卻困難得多。例如,考慮的案例為多面體的矩陣,也就是說,在這里M1到Mr是給定的。與此相對(duì)照的情況涉及多項(xiàng)式,這體現(xiàn)為文[8]中,認(rèn)為它并不足以檢查子集這包含多面體的各個(gè)二維面。此外，從[8]中可以看出即使每個(gè)矩陣Mi 包含一個(gè)非零元素，檢查 都不合格，也就是說超矩形的情況。一個(gè)設(shè)定的滿足的結(jié)果在[22]中。說明這是顯示,它充分檢查每個(gè)點(diǎn)上為了說明在 的組成元素有特殊結(jié)構(gòu)時(shí)改進(jìn)是有可能的，考慮這里In1指的是（n1)階矩陣。在這個(gè)例子中,它充分檢查設(shè)置它包含四個(gè)矩陣?！坝?)給定的是足夠的”是Kharitonov結(jié)論的一個(gè)結(jié)果，因?yàn)槊總€(gè)矩陣都是相似的形式。不管n等于多少， 都包含四個(gè)矩陣。所以，當(dāng)有特殊結(jié)構(gòu)的時(shí)候，簡(jiǎn)化是可能的。一個(gè)直接的依賴證明這一點(diǎn)的結(jié)果是未知的。這樣可能會(huì)導(dǎo)致改善證明polytopes治療更一般的矩陣。這個(gè)矩陣指數(shù)扮演著重要的角色,在線性系統(tǒng)和控制原理。在這里我們予以審查過程中所扮演的角色矩陣指數(shù),更有一些它有趣的屬性,并提出一些相關(guān)的問題尚未解決?？紤]這個(gè)線性系統(tǒng)：在這里狀態(tài)變量，控制變量 。狀態(tài)變量 x(t) 由以下著名公式清楚地給出如果x0= 0 且允許 u(.) 是一個(gè)在[0, t)任意可積的函數(shù), 那么所有狀態(tài)變量x(t) 在時(shí)間t是Rn代碼所給出的子空間的半正定矩陣定義的范圍內(nèi)另外, 因?yàn)?t 0 時(shí)，Q(t) 的變化與t無關(guān)而是由 [18, 56, 74, 951]給出這里 I