In the modern USA, some politicians have declared a metaphorical war on crime. The US federal and state governments have far more military power than all the criminals in the USA put together, let alone any one criminal organization.
It is certainly possible for governments to adopt policies which will greatly reduce the crime rate in the future. But any "war on crime", or any other approach to reducing crime rates, is doomed to gone on forever without total success. Because the root cause of crime in the "evil that lurks in the hearts of men" (and in the hearts of women and children too). Government policies can greatly reduce the evil impulses lurking in the hearts of humans, and/or the rate at which humans follow those evil impulses, but it seems very improbable that any policies can totally eliminate crime.
So a metaphorical war on crime can drag on forever, despite victory after victory after victory over individual criminals and criminal groups, since the only two ways to end a metaphorical war on crime are: One) to give up, abolish all laws, and make all actions, no matter how evil, legal, or else Two) to lie and declare victory over crime, despite crimes still happening and criminals still being punished for those crimes.
In your case the wars are not exactly metaphorical wars on crime, but more like real wars or like civil wars, since I am not certain about the exact political relationship between the central government and the local governments in the outer regions. Are the outer states more like largely autonomous provinces or tributary states or vassal kingdoms, or are the outer states more like foreign countries, or are they somewhere in between? Somewhere in between being a dependent state and an independent state was a very common political condition for most of recorded history.
Anyway, it seems like the conflicts in the outer regions are a bit too large to be a metaphorical war on crime in the other regions, and more like foreign wars or civil wars.
However, the public justifications, though not necessarily the reasons, for various attacks on various outer regions could be various crimes and sins committed by their rulers. One of the functions of the central government could be to guarantee moral and ethical behavior standards by the rulers of states in the outer regions. So whenever the central imperial government, or an ambitious governor of a province, wants to attack one of the outer states, they can use the list of misdeeds of the ruler of that state which have not yet been punished as justification for the attack.
So the conflicts in the outer regions seem a lot more like wars than like crime.
But wars vary a lot in severity. On one extreme there is World War Two and the long feared but as yet not happened World War Three, causing horror and suffering, death and destruction, on a vast and unimaginable scale.
And on the other extreme of the scale is endemic low intensity warfare where each small tribe or band of people is in a perpetual state of war with all their neighbors, but acts of war are few and tiny in scope.
For example, before Europeans arrived there was no central government in the vast area of the trans Mississippi west in the USA. There were hundreds of independent nations, tribes, bands, and other groups of people, each with the ability to make war on other groups. Some of those groups had warlike cultures where young men were expected to make their reputations and gain wealth by killing and stealing from members of other groups. So each of those warlike groups was in a constant state of war with all their neighbors, except for any groups which might be allied to them.
So if you assume that the average total population in the trans Mississippi west was between 100,000 and 500,000 and assume that the average fatality rate in that low level endemic warfare was between a tenth of a percent and one percent, about 100 to 5,000 men, women, and children would die in that warfare in the average year.
The USA kept at least a few troops stationed in various forts in the trans Mississippi west to maintain peace with various Indian groups there for over a century. And there were occasional conflicts with various Indian groups there over most of that period. But the period of really intense warfare with various hostile groups there only lasted about 50 years from 1850 to 1900, or only 30 years from 1850 to 1880, or only 20 years from about 1857 to 1877, or various other relatively short lengths of time depending on when it was defined as beginning and ending. It only took about a generation to pacify the most hostile groups in the west, so that is not a good example of warfare lasting a long time.
But the USA did have various Indian wars with various groups in the eastern USA on and off from the adoption of the Constitution in 1789 to the end of the Third Seminole War in 1858, somewhat overlapping with the western Indian Wars. And the USA fought various major wars with foreign powers such as the War of 1812, the Mexican-American War, the Civil War, the Spanish-American War, World War One, World War Two, the Korean War, the Vietnam War, the Gulf War, the Wars in Afghanistan and Iraq, etc.
And the USA has had many other small conflicts such as Barbary Pirates Wars, the Mormon War, the Korean Expedition in 1871, intervention in various Latin American countries, etc., so it has become a great and successful nation despite, and sometimes because, of being in an almost constant state of at least low intensity War.
Similarly the United Kingdom flourished in the 19th century despite fighting such major wars as the Napoleonic Wars, the Crimean War, The Indian Rebellion, and the Boer War, as well as being constantly involved in at least one small scale colonial war somewhere in the world.
So if you imagine that the central power in your story resembles in someways the 19th century USA or the 19th century UK, the wars in the outer region should not drain its power or weaken it and it could go on fighting wars at that level of intensity forever.
But what about the outer groups that the central power keeps fighting? How long can they keep on fighting occasional wars against it before they are destroyed or pacified? In the US west it was very hard to find an Indian tribe that would fight repeated wars against the USA over a long period of time.
The Chiricahua Apaches were mostly at peace with Americans from the Mexican War in 1846-48 to 1861, then at war until 1872, then at peace until about 1876, then alternately at war and peace on and off until 1886. So basically the Chiricahuas were at war on and off over a period about 25 years. The First Sioux War was in 1854-56 and the Ghost Dance Troubles were in 1890-91, so various plains Sioux were only hostile on and off for 37 years.
As a rule Indian tribes didn't keep repeating the mistake of fighting the USA over and over again.
The United Kingdom mostly defeated its colonial enemies in its wars and added them to the British Empire. The longest lasting colonial enemy of Britain might have been Afghanistan, which fought the first, Second, and Third Anglo-Afghan Wars in 1839-42, 1878-80, and 1919, and remained an independent country at the end of them, after a period of 80 years.
On the other hand, the Spanish government claimed to rule what is now the Southwestern USA for about 200 years, and during that period there were repeated conflicts, on and off, with various groups of Navajos, Apaches, and Comanches, as well as other conflicts with other Indian groups. There were times when the Spanish settlers were suffering greatly from hostile raids, and other times when the Spanish authorities succeeded in achieving almost total peace with almost all groups. As far as I can tell the pattern of repeated on and off warfare could have gone on forever without destroying or permanently pacifying the Indians.
In ancient and medieval history there are examples of conditions of endemic low intensity on and off warfare seeming to go on for decades and centuries without any resolution.
For centuries there were dozens of small city states in ancient Greece, for example, which sometimes went to war with other city states. At any moment someone could face an invasion from a neighboring city state which might be only a single day's march away. But even in such uncertain conditions many of the Greek city states were prosperous for 500 or 1,000 years.
In medieval western Europe for centuries the feudal system enabled even the lowest ranking lords to fight private wars against other low ranking lords. But even under those conditions western Europe gradually became more prosperous, more populated, and more civilized.
So it seems that a society can exist in a state of endemic on and off warfare, when that warfare is not severe enough or frequent enough to reach a level of causing it to decline.
The more severe the conflicts, the rarer they must be for the society to survive without major damage. The more frequent the conflicts, the less severe they must be for the society not to be fatally damaged by them.