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"The Choctaw Civil War was a period of economic and social unrest among the Choctaw people that degenerated into a civil war between 1747 and 1750. The war was fought between two different factions within the Choctaw over what the tribes's trade relations with Great Britain and France should be. Hundreds of Choctaw peoples died in the war and the pro-French faction retained their influence within the Choctaw nation. Background Residing in Mississippi, Louisiana, and Alabama, by the early 18th century the Choctaw people were under threat from a number of regional rivals. The Choctaw warred intermittently with the Chickasaw to the north, while the Muscogee-Creek frequently raided the western Choctaw territories, with both tribes enslaving thousands of Choctaws for use as laborers in the West Indies.Brescia, William (Bill) (1982). "Chapter 2, French-Choctaw Contact, 1680s–1763". Tribal Government, A New Era. Philadelphia, Mississippi: Choctaw Heritage Press.O’BRIEN, GREG. "Quieting the Ghosts: How the Choctaws and Chickasaws Stopped Fighting." In The Native South: New Histories and Enduring Legacies, edited by Garrison Tim Alan and O’Brien Greg, 47-69. LINCOLN; LONDON: University of Nebraska Press, 2017. Accessed April 29, 2020. www.jstor.org/stable/j.ctt1q1xq7h.7. These rival tribes were armed with British-provided firearms, often bought using proceeds from slave raids.GALLAY, ALAN. "CONTOURS OF THE INDIAN SLAVE TRADE." In The Indian Slave Trade: The Rise of the English Empire in the American South, 1670–1717, 288-314. Yale University Press, 2002. Accessed April 30, 2020. www.jstor.org/stable/j.ctt1npmvw.20. Faced with these threats, the Choctaw themselves began to militarize, and, using money brought in from the growing fur trade, began to buy firearms from French colonists who had settled in Louisiana.Struggle and Survival in Colonial America By David G. Sweet, Gary B. Nash Edition: illustrated Published by University of California Press, 1982 , . Pp. 49–67. The Choctaw also organized themselves into rough political units described as "divisions". Two divisions eventually rose to dominate Choctaw politics, the Eastern Division (the Okla Tannap, or People of the Opposite Side) and the Western Division (the Okla Falaya, the Long People). A third, smaller division (the Okla Hannali, the People of Sixtowns) known as the Southern Division also existed, but its members often isolated themselves from the affairs of the other two divisions or remained neutral. Friendly relations between the Choctaw and the French colonists of Louisiana continued into the 1730s; the Choctaw saw the French as a valuable economic partner (trading furs for guns, offering guides for hire, and as a market for enslaved prisoners), while the French viewed their alliance with the Choctaw as vital to the defense of lower Louisiana from Britain and its Chickasaw allies. British merchants attempted to establish trade relations with the Eastern Division in 1731, but this effort failed when another war broke out between the Choctaw and the Chickasaw later that year. Despite this failure, the French feared that Britain was attempting to destabilize the French alliance with the Choctaw. As such, the French greatly strengthened their military ties to the Choctaw, building Fort Tombecbe in Choctaw territory and conducting a joint Choctaw–French expedition to destroy Chickasaw villages on the Tombigbee river in 1736. By the late 1730s the Choctaw military situation had stabilized (fighting would continue against the remaining Chickasaw into the 1760s),Atkinson, James R. (2004). Splendid Land, Splendid People. University of Alabama Press. . pp. 25–87 and the fur trade continued to increase the wealth of the Choctaw people. The various tribes remained on friendly terms with the French, while the British were tolerated. By 1740, the Choctaw numbered some 12,000 people living in around 50 villages. Civil War During the wars with the Chickasaw, a Choctaw warrior known as Red Shoes rose to prominence. Although he was not born into one of the hereditary chiefdoms that made up the Choctaw nation, Red Shoes was able to rise to the position of war-captain due to his war-won fame. He eventually became chief of the town of Couechitto, which soon grew wealthy due to its involvement with the fur trade. However, after several years of lackluster trading with the French, Red Shoes decided to enter trade negotiations with the British. Red Shoe's trading with the British threatened to damage the tribe's relations with the French, and when it was discovered the chief was widely chastised by other Choctaw leaders. However, Red Shoes continued to trade with the British, even going so far as to lead a delegation to Charleston.Foret, Michael J. "War or Peace? Louisiana, the Choctaws, and the Chickasaws, 1733-1735." Louisiana History: The Journal of the Louisiana Historical Association 31, no. 3 (1990): 273-92. Accessed April 29, 2020. www.jstor.org/stable/4232807. Red Shoes found that the British were willing to sell guns and other goods to the Choctaw at lower prices than the French were, and soon the chief (who was affiliated with the Western Division) was able to further increase his political power. Though he had fought alongside the French against the Chickasaw, his observation of the French Tombigbee expedition (which had suffered heavy casualties) convinced him that the Louisiana colony was militarily weak. He also considered allying his large tribe with British colonists in the Carolinas. In addition, Red Shoes sought to enter peace negotiations with the Chickasaw, an idea which many Choctaw and French leaders strongly opposed.USNER, DANIEL H. "THE INDIAN ALLIANCE NETWORK OF A MARGINAL EUROPEAN COLONY." In Indians, Settlers, and Slaves in a Frontier Exchange Economy: The Lower Mississippi Valley Before 1783, 77-104. CHAPEL HILL; LONDON: University of North Carolina Press, 1992. Accessed April 29, 2020. doi:10.5149/9780807839966_usner.9. Red Shoes' shift in policy and growing political power caused political discord within the Choctaw, and by the late 1730s tensions were rising between the eastern and western divisions.Galloway, Patricia. "THE BARTHELEMY MURDERS: BIENVILLE'S ESTABLISHMENT OF THE "LEX TALIONIS" AS A PRINCIPLE OF INDIAN DIPLOMACY." Proceedings of the Meeting of the French Colonial Historical Society 8 (1985): 91-103. Accessed April 29, 2020. www.jstor.org/stable/42952133. Small-scale fighting between Choctaw villages began in November 1739, and while this was quickly stopped through inter-tribal mediation it showed the increasing divides in Choctaw society. 1742 was a bloody year in the war against the Chickasaw, and a smallpox outbreak in the summer of that year killed many Choctaw children and elders. Fields went unattended - many farmers having been recruited for war parties - leading to a food shortage, and the movement of Choctaw warriors throughout the people's territory helped to spread disease. Further exacerbating these issues, in 1745 a newly-appointed French governor of Louisiana - seeking to cut colony expenses - decreased the annual gifts given to the Choctaw leaders and imposed new trading restrictions on French manufactured goods. These changes harmed the longstanding Choctaw-French trade, and further convinced the Western Division that the British would make better trading partners. In 1746, a trio of French traders in Couechitto were captured and executed (possibly in retaliation for a rape) by warriors loyal to Red Shoes. Soon after, the Western Division expelled all traders from their territory. This development greatly angered the Eastern Division and their French allies, and leaders from the two groups soon met to discuss military action against the Western Division. The French also sent envoys to the Western Division, but were told by pro-French chiefs that they lacked the power to destroy Red Shoes and his pro-British supporters. With war imminent, the French conspired to assassinate the Red Shoes, offering a large bounty on him. On the night of 23 June 1747 a young Choctaw warrior acting as an escort for a British merchant assassinated Red Shoes in his sleep, plunging Red Shoe's tribe into chaos. His faction did not dissolve, however, and when French complicity with the assassination was later discovered many tribes affiliated with the Western Division demanded revenge. The situation was further inflamed by the French, who demanded the deaths of two additional Western Division chiefs as recompense for the three slain Frenchmen. The eastern and western divisions did not attack each-other immediately, instead choosing to target their respective allies, but when a western-aligned chief was killed in an eastern raid on a British caravan, open war broke out. The first major action of the war came in July of 1748 when the Eastern Division launched a raid on Couechitto, Red Chief's former village. The war itself was bloody and chaotic, with both sides attacking each other's villages and towns. Raids were common, and due to the dispersed nature of Choctaw settlements, entire villages were often slaughtered before their respective division could arrive to aid them. Hundreds of Choctaws were killed. Both sides were armed by their respective European allies, with the British backing the Western Division and the French supporting the Eastern Division. Using Fort Tombecbe as a base, the French sent colonial troops to aid their eastern allies in battle, offering three times the pre-war price for enemy scalps. The Eastern Division possessed a numerical advantage over the Western Division, and this allowed the former faction to wear down their opponent through a brutal war of attrition. Benefiting from their pre-war diplomacy, the Western Division allied themselves with the Chakchiuma and the Chickasaw. With British support, the Western Division launched an unsuccessful attack on the key eastern- aligned town of Oulitacha that cost the lives of over 100 warriors from both sides. The Western Division won a few minor victories, but by 1750 a number of devastating raids had sapped the ability of the Western Division to wage war, and the disruption caused to the fur trade by the conflict had collapsed the economies of both factions. The Western Division was also running critically low on supplies, as the end of King George's War (1744–1748) had reduced the British need to destabilize French Louisiana; according to one anecdote given by a British trader, the Western Division was so short on ammunition they were resorting to firing glass beads. Faced with defeat, the Western Division signed a peace treaty in November 1750. The surviving Choctaw leaders concluded a treaty favorable to the Eastern Division and the French. Under the terms of said treaty, the Choctaw promised to avenge the death of any Frenchmen with the death of a rebel Choctaw, kill any remaining British merchants and their Choctaw sponsors, and continue the war against the Chickasaw. The Western Division was also forced to destroy fortified settlements in their territory and exchange prisoners. = Aftermath = The war left hundreds of Choctaws dead. Dozens of villages were destroyed, and many more were damaged or depopulated. With the Eastern Division established as the dominant force among the Choctaw people, the faction ended trade with Britain and returned to the pre-war trade with the French. However, the war also caused the Choctaw to re-evaluate their relations with the European powers; in the postwar years, the Choctaw adopted a flexible, pragmatic stance towards Britain, France, and Spain, eventually resulting the Choctaw supporting Britain in the Seven Years' War and both Britain and Spain during the American Revolutionary War. References Civil wars involving the states and peoples of North America Civil wars of the Early Modern era Choctaw Conflicts in 1747 Conflicts in 1748 Conflicts in 1749 Conflicts in 1750 1747 in North America 1748 in North America 1749 in North America 1750 in North America "
"The separation principle is one of the fundamental principles of stochastic control theory, which states that the problems of optimal control and state estimation can be decoupled under certain conditions. In its most basic formulation it deals with a linear stochastic system :begin{align} dx & =A(t)x(t),dt+B_1(t)u(t),dt+B_2(t),dw dy & =C(t)x(t),dt +D(t),dw end{align} with a state process x, an output process y and a control u, where w is a vector-valued Wiener process, x(0) is a zero-mean Gaussian random vector independent of w, y(0)=0, and A, B_1, B_2, C, D are matrix-valued functions which generally are taken to be continuous of bounded variation. Moreover, DD' is nonsingular on some interval [0,T]. The problem is to design an output feedback law pi:, y mapsto u which maps the observed process y to the control input u in a nonanticipatory manner so as to minimize the functional : J(u) = mathbb{E}left{ int_0^T x(t)'Q(t)x(t),dt+int_0^Tu(t)'R(t)u(t),dt +x(T)'Sx(T)right}, where mathbb{E} denotes expected value, prime (') denotes transpose. and Q and R are continuous matrix functions of bounded variation, Q(t) is positive semi- definite and R(t) is positive definite for all t. Under suitable conditions, which need to be properly stated, the optimal policy pi can be chosen in the form : u(t)=K(t)hat x(t), where hat x(t) is the linear least-squares estimate of the state vector x(t) obtained from the Kalman filter : dhat x=A(t)hat x(t),dt+B_1(t)u(t),dt +L(t)(dy-C(t)hat x(t),dt),quad hat x(0)=0, where K is the gain of the optimal linear-quadratic regulator obtained by taking B_2=D=0 and x(0) deterministic, and where L is the Kalman gain. There is also a non-Gaussian version of this problem (to be discussed below) where the Wiener process w is replaced by a more general square- integrable martingale with possible jumps.. In this case, the Kalman filter needs to be replace by a nonlinear filter providing an estimate of the (strict sense) conditional mean : hat{x}(t)= operatorname E{ x(t)mid {cal Y}_t}, where : {cal Y}_t:=sigma{ y(tau), tauin [0,t]}, quad 0leq tleq T, is the filtration generated by the output process; i.e., the family of increasing sigma fields representing the data as it is produced. In the early literature on the separation principle it was common to allow as admissible controls u all processes that are adapted to the filtration {{cal Y}_t, , 0leq tleq T}. This is equivalent to allowing all non- anticipatory Borel functions as feedback laws, which raises the question of existence of a unique solution to the equations of the feedback loop. Moreover, one needs to exclude the possibility that a nonlinear controller extracts more information from the data than what is possible with a linear control law.. Choices of the class of admissible control laws Linear- quadratic control problems are often solved by a completion-of-squares argument. In our present context we have : J(u)=operatorname{E}left{ int_0^T(u-Kx)'R(u-Kx) , dtright} +text{terms that do not depend on }u, in which the first term takes the form. :begin{align} operatorname{E}left{ int_0^T(u-Kx)'R(u-Kx),dtright}=operatorname{E}left{int_0^T[(u-Khat{x})'R(u-Khat{x})+operatorname{tr}(K'RKSigma)] , dtright}, end{align} where Sigma is the covariance matrix : Sigma(t):=operatorname{E}{[x(t)-hat{x}(t)][x(t)-hat{x}(t)]'}. The separation principle would now follow immediately if begin{align}Sigmaend{align} were independent of the control. However this needs to be established. The state equation can be integrated to take the form : x(t)=x_0(t)+int_0^t Phi(t,s)B_1(s)u(s) , ds, where x_0 is the state process obtained by setting u=0 and Phi is the transition matrix function. By linearity, hat{x}(t)=operatorname{E}{x(t)mid {cal Y}_t} equals : hat{x}(t)=hat{x}_0(t)+int_0^t Phi(t,s)B_1(s)u(s),ds, where hat{x}_0(t)=operatorname{E}{x_0(t)mid {cal Y}_t}. Consequently, : Sigma(t):=mathbb{E}{[x_0(t)-hat{x}_0(t)][x_0(t)-hat{x}_0(t)]'}, but we need to establish that begin{align}hat{x}_0end{align} does not depend on the control. This would be the case if : {cal Y}_t ={cal Y}_t^0:=sigma{ y_0(tau), tauin [0,t]}, quad 0leq tleq T, where y_0 is the output process obtained by setting u=0. This issue was discussed in detail by Lindquist. In fact, since the control process u is in general a nonlinear function of the data and thus non-Gaussian, then so is the output process y. To avoid these problems one might begin by uncoupling the feedback loop and determine an optimal control process in the class of stochastic processes u that are adapted to the family { {cal Y}_t^0} of sigma fields. This problem, where one optimizes over the class of all control processes adapted to a fixed filtration, is called a stochastic open loop (SOL) problem. It is not uncommon in the literature to assume from the outset that the control is adapted to { {mathcal Y}_t^0}; see, e.g., Section 2.3 in Bensoussan,. also van Handel and Willems.. In Lindquist 1973 a procedure was proposed for how to embed the class of admissible controls in various SOL classes in a problem-dependent manner, and then construct the corresponding feedback law. The largest class Pi of admissible feedback laws pi consists of the non- anticipatory functions u:=pi(y) such that the feedback equation has a unique solution and the corresponding control process u_pi is adapted to {{mathcal Y}_t^0}. Next, we give a few examples of specific classes of feedback laws that belong to this general class, as well as some other strategies in the literature to overcome the problems described above. =Linear control laws= The admissible class Pi of control laws could be restricted to contain only certain linear ones as in Davis.. More generally, the linear class : ({mathcal L})quad u(t)=bar{u}(t)+int_0^tF(t,tau),dy, where bar{u} is a deterministic function and F is an L_2 kernel, ensures that Sigma is independent of the control. In fact, the Gaussian property will then be preserved, and hat{x} will be generated by the Kalman filter. Then the error process tilde{x}:= x-hat{x} is generated by : dtilde{x}=(A-LC)tilde{x},dt +(B_2-LD),dw, quad tilde{x}(0)=x(0), which is clearly independent of the choice of control, and thus so is Sigma. =Lipschitz-continuous control laws= Wonham proved a separation theorem for controls in the class begin{align}pi:, u(t)=psi(t,hat{x}(t))end{align}, even for a more general cost functional than J(u). However, the proof is far from simple and there are many technical assumptions. For example, begin{align}C(t)end{align} must square and have a determinant bounded away from zero, which is a serious restriction. A later proof by Fleming and Rishel. is considerably simpler. They also prove the separation theorem with quadratic cost functional J(u) for a class of Lipschitz continuous feedback laws, namely u(t)=phi(t,y), where phi:, [0,T]times C^n [0,T]to{mathbb R}^m is a non-anticipatory function of y which is Lipschitz continuous in this argument. Kushner. proposed a more restricted class u(t)=psi(t,hat{xi}(t)), where the modified state process hat{xi} is given by : hat{xi}(t)=operatorname{E}{ x_0(t)mid {mathcal Y}_t^0}+ int_0^t Phi(t,s)B_1(s)u(s),ds, leading to the identity begin{align}hat{x}=hat{xi}end{align}. =Imposing delay= If there is a delay in the processing of the observed data so that, for each t, u(t) is a function of y(tau); , 0leqtauleq t-varepsilon, then {cal Y}_t ={cal Y}_t^0, 0leq tleq T, see Example 3 in Georgiou and Lindquist. Consequently, Sigma is independent of the control. Nevertheless, the control policy pi must be such that the feedback equations have a unique solution. Consequently, the problem with possibly control-dependent sigma fields does not occur in the usual discrete-time formulation. However, a procedure used in several textbooks to construct the continuous-time Sigma as the limit of finite difference quotients of the discrete-time Sigma, which does not depend on the control, is circular or a best incomplete; see Remark 4 in Georgiou and Lindquist. =Weak solutions= An approach introduced by Duncan and Varaiya. and Davis and Varaiya,. see also Section 2.4 in Bensoussan is based on weak solutions of the stochastic differential equation. Considering such solutions of : dx =A(t)x(t),dt+B_1(t)u(t),dt+B_2(t),dw we can change the probability measure (that depends on begin{align}uend{align}) via a Girsanov transformation so that : dtilde{w}:= B_1(t)u(t),dt+B_2(t),dw becomes a new Wiener process, which (under the new probability measure) can be assumed to be unaffected by the control. The question of how this could be implemented in an engineering system is left open. =Nonlinear filtering solutions= Although a nonlinear control law will produce a non-Gaussian state process, it can be shown, using nonlinear filtering theory (Chapters 16.1 in Lipster and Shirayev. ), that the state process is conditionally Gaussian given the filtration begin{align}{{mathcal Y}_t}end{align}. This fact can be used to show that begin{align}hat{x}end{align} is actually generated by a Kalman filter (see Chapters 11 and 12 in Lipster and Shirayev). However, this requires quite a sophisticated analysis and is restricted to the case where the driving noise begin{align}wend{align} is a Wiener process. Additional historical perspective can be found in Mitter.. Issues on feedback in linear stochastic systems At this point it is suitable to consider a more general class of controlled linear stochastic systems that also covers systems with time delays, namely :begin{align} z(t) & =z_0(t) + int_0^t G(t,s)u(s),ds y(t) & = Hz(t) end{align} with begin{align}z_0end{align} a stochastic vector process which does not depend on the control. The standard stochastic system is then obtained as a special case where z=[x',y']', z_0=[x_0',y_0']' and H=[I,0]. We shall use the short-hand notation : z=z_0+gpi Hz for the feedback system, where : g;:; (t,u) mapsto int_0^t G(t,tau)u(tau),dtau is a Volterra operator. In this more general formulation the embedding procedure of Lindquist defines the class Pi of admissible feedback laws pi as the class of non-anticipatory functions u:=pi(y) such that the feedback equation z=z_0+gpi Hz has a unique solution z_pi and u=pi(Hz_pi) is adapted to {{mathcal Y}_t^0}. In Georgiou and Lindquist a new framework for the separation principle was proposed. This approach considers stochastic systems as well-defined maps between sample paths rather than between stochastic processes and allows us to extend the separation principle to systems driven by martingales with possible jumps. The approach is motivated by engineering thinking where systems and feedback loops process signals, and not stochastic processes per se or transformations of probability measures. Hence the purpose is to create a natural class of admissible control laws that make engineering sense, including those that are nonlinear and discontinuous. The feedback equation z=z_0+gpi Hz has a unique strong solution if there exists a non-anticipating function F such that z=F(z_0) satisfies the equation with probability one and all other solutions coincide with z with probability one. However, in the sample-wise setting, more is required, namely that such a unique solution exists and that z=z_0+gpi Hz holds for all z_0, not just almost all. The resulting feedback loop is deterministically well-posedin the sense that the feedback equations admit a unique solution that causally depends on the input for each input sample path. In this context, a signal is defined to be a sample path of a stochastic process with possible discontinuities. More precisely, signals will belong to the Skorohod space D, i.e., the space of functions which are continuous on the right and have a left limit at all points (càdlàg functions). In particular, the space C of continuous functions is a proper subspace of D. Hence the response of a typical nonlinear operation that involves thresholding and switching can be modeled as a signal. The same goes for sample paths of counting processes and other martingales. A system is defined to be a measurable non-anticipatory map Dto D sending sample paths to sample paths so that their outputs at any time t is a measurable function of past values of the input and time. For example, stochastic differential equations with Lipschitz coefficients driven by a Wiener process induce maps between corresponding path spaces, see page 127 in Rogers and Williams,. and pages 126-128 in Klebaner.. Also, under fairly general conditions (see e.g., Chapter V in Protter.), stochastic differential equations driven by martingales with sample paths in D have strong solutions who are semi-martingales. For the time setting f(z):=gpi Hz, the feedback system z=z_0+gpi Hz can be written z=z_0+f(z), where z_0 can be interpreted as an input. Definition. A feedback loop z=z_0+f(z) is deterministically well- posed if it has a unique solution zin D for all inputs z_0in D and (1-f)^{-1} is a system. This implies that the processes z and z_0 define identical filtrations. Consequently, no new information is created by the loop. However, what we need is that {cal Y}_t ={cal Y}_t^0 for 0leq tleq T. This is ensured by the following lemma (Lemma 8 in Georgiou and Lindquist). Key Lemma. If the feedback loop z=z_0+gpi Hz is deterministically well-posed, gpi is a system, and H is a linear system having a right inverse H^{-R} that is also a system, then (1-Hgpi)^{-1} is a system and {cal Y}_t ={cal Y}_t^0 for 0leq tleq T. The condition on H in this lemma is clearly satisfied in the standard linear stochastic system, for which H=[0,I], and hence H^{-R}=H'. The remauining conditions are collected in the following definition. Definition. A feedback law pi is deterministically well-posed for the system z=z_0+gpi Hz if gpi is a system and the feedback system z=z_0+gpi Hz deterministically well-posed. Examples of simple systems that are not deterministically well-posed are given in Remark 12 in Georgiou and Lindquist. A separation principle for physically realizable control laws By only considering feedback laws that are deterministically well-posed, all admissible control laws are physically realizable in the engineering sense that they induce a signal that travels through the feedback loop. The proof of the following theorem can be found in Georgiou and Lindquist 2013. Separation theorem. Given the linear stochastic system : begin{align} dx & =A(t)x(t),dt+B_1(t)u(t),dt+B_2(t),dw dy & =C(t)x(t),dt +D(t),dw end{align} where w is a vector-valued Wiener process, x(0) is a zero-mean Gaussian random vector independent of w, consider the problem of minimizing the quadratic functional J(u) over the class of all deterministically well- posed feedback laws pi. Then the unique optimal control law is given by u(t)=K(t)hat{x}(t) where K is defined as above and hat{x} is given by the Kalman filter. More generally, if w is a square-integrable martingale and x(0) is an arbitrary zero mean random vector, u(t)=K(t)hat{x}(t), where hat{x}(t)=operatorname{E}{x(t)mid {cal Y}_t}, is the optimal control law provided it is deterministically well-posed. In the general non-Gaussian case, which may involve counting processes, the Kalman filter needs to be replaced by an nonlinear filter. A Separation principle for delay- differential systems Stochastic control for time-delay systems were first studied in Lindquist,.. and Brooks,. although Brooks relies on the strong assumption that the observation y is functionally independent of the control u, thus avoiding the key question of feedback. Consider the delay-differential system :begin{align} dx &=left(int_{t-h}^t d_s,A(t,s)x(s)right) ,dt + B_1(t)u(t),dt+B_2(t),dw dy & =left(int_{t-h}^t d_s,C(t,s)x(s)right) ,dt +D(t),dw end{align} where w is now a (square-integrable) Gaussian (vector) martingale, and where begin{align}Aend{align} and C are of bounded variation in the first argument and continuous on the right in the second, x(t)=xi(t) is deterministic for -hleq tleq 0, and y(0)=0. More precisely, A(t,s)=0 for sgeq t, A(t,s)=A(t,t-h) for tleq t-h, and the total variation of smapsto A(t,s) is bounded by an integrable function in the variable t, and the same holds for C. We want to determine a control law which minimizes : J(u)=operatorname{E}left(int_0^T x(t)'Q(t)x(t),dalpha(t)+int_0^Tu(t)'R(t)u(t),dtright), where begin{align}dalphaend{align} is a positive Stieltjes measure. The corresponding deterministic problem obtained by setting begin{align}w=0end{align} is given by : u(t)=int_{t-h}^t d_tau , K(t,tau)x(tau), with begin{align}Kend{align}. The following separation principle for the delay system above can be found in Georgiou and Lindquist 2013 and generalizes the corresponding result in Lindquist 1973. Theorem. There is a unique feedback law begin{align}pi:, ymapsto uend{align} in the class of deterministically well-posed control laws that minimizes begin{align}J(u)end{align}, and it is given by : u(t)=int_{t-h}^t d_s , K(t,s)hat{x}(smid t), where K is the deterministic control gain and hat{x}(smid t) := E{ x(s)mid {cal Y}_t} is given by the linear (distributed) filter :begin{align} dhat{x}(tmid t) & =int_{t-h}^t d_s , A(t,s)hat{x}(smid t) , dt +B_1u,dt+ X(t,t),dv dhat{x}(tmid t) & =int_{t-h}^t d_s , A(t,s)hat{x}(smid t) , dt +B_1u,dt+ X(t,t),dv end{align} where v is the innovation process : dv=dy - int_{t-h}^t d_sC(t,s)hat{x}(smid t), dt, quad v(0)=0, and the gain x is as defined in page 120 in Lindquist. References Control theory Stochastic control "
"The Espolin Gallery () is an art gallery in Storvågan near Kabelvåg in the municipality of Vågan, Norway. The gallery was established in 1992 to display the works of the artist Kaare Espolin Johnson. In addition to Espolin Johnson's works, the gallery also displays works by other artists. The gallery is now part of the SKREI Heritage Center (SKREI Opplevelsessenter) at Museum Nord. Architecture The building was designed by the architect Gisle Jakhelln. He took his inspiration from traditional Icelandic construction because Espolin Johnson had Icelandic ancestors. In June 2006, the gallery was expanded with a new wing measuring around . The gallery received financial support for the expansion from the Arts Council Norway, Nordland County Council, the municipality of Vågan, and the bank Sparebanken Nord-Norge. References External links *The Espolin Gallery at the Museum Nord website *The Espolins Gallery website Art museums and galleries in Norway Culture in Nordland Vågan Art museums established in 1992 "